The Hitchhiker guide to: Secant Varieties and Tensor Decomposition
Alessandra Bernardi Enrico Carlini Maria Virginia Catalisano Alessandro Gimigliano Alessandro Oneto




A. Bernardi]Alessandra Bernardi^*
[A. Bernardi]Dipartimento di Matematica, Università di Trento, Trento, Italy
alessandra.bernardi@unitn.it

E. Carlini]Enrico Carlini^*
[E. Carlini]Dipartimento di Scienze Matematiche, Politecnico di Torino, Turin, Italy
enrico.carlini@polito.it

M.V. Catalisano]Maria Virginia Catalisano^*
[M.V. Catalisano]Dipartimento di Ingegneria Meccanica, Energetica, Gestionale e dei Trasporti, Università degli studi di Genova, Genoa, Italy,
catalisano@dime.unige.it

A. Gimigliano]Alessandro Gimigliano^*
[A. Gimigliano]Dipartimento di Matematica, Univ. di Bologna, Bologna, Italy
Alessandr.Gimigliano@unibo.it

A. Oneto]Alessandro Oneto^*
[A. Oneto]Barcelona Graduate School of Mathematics, and Universitat Politècnica de Catalunya, Barcelona, Spain
alessandro.oneto@upc.edu; aless.oneto@gmail.com

[2010]13D40, 14A05, 14A10, 14A15, 14C20, 14M12, 14M15, 14M99, 14N05, 15A69, 15A75

The Hitchhiker guide to: Secant Varieties and Tensor Decomposition.
    [
    December 30, 2023
===================================================================



^* The initial elaboration of this work had Anthony V. Geramita

as one of the authors. The paper is dedicated to him.




We consider here the problem, which is quite classical in Algebraic
geometry, of studying the secant varieties of a projective variety
X. The case we concentrate on is when X is a Veronese variety, a
Grassmannian or a Segre variety. Not only these varieties are among
the ones that have been most classically studied, but a strong
motivation in taking them into consideration is the fact that they
parameterize, respectively, symmetric, skew-symmetric and general
tensors, which are decomposable, and their secant varieties give a
stratification of tensors via tensor rank. We collect here most of
the known results and the open problems on this fascinating subject.


Keywords. Additive decompositions; Secant varieties; Veronese varieties; Segre varieties; Segre-Veronese varieties; Grassmannians; tensor rank; Waring rank; algorithm.





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 §.§ Funding

	A.B. acknowledges the Narodowe Centrum Nauki, Warsaw Center of Mathematics and Computer Science, Institute of Mathematics of university of Warsaw which financed  the 36th Autumn School in Algebraic Geometry “Power sum decompositions and apolarity, a geometric approach" September 1st-7th, 2013 Łukęcin, Poland where a partial version of the present paper was conceived. M.V. C. and A. G. acknowledges financial support from MIUR (Italy). A.O. acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445).
	

toc






§ INTRODUCTION






 §.§ The Classical Problem


When considering finite dimensional vector spaces over a field
 (which for us, will always be algebraically closed and of
characteristic zero, unless stated otherwise), there are three main functors
that come to attention when doing multilinear algebra:

	
  * the tensor product, denoted by V_1⊗⋯⊗ V_d;
	
  * the symmetric product, denoted by S^dV; 
	
  * the wedge product, denoted by ⋀^dV.


These functors are associated with three classically-studied projective varieties in algebraic geometry (see e.g., <cit.>): 

	
  * the Segre variety;
	
  * the Veronese variety;
	
  * the Grassmannian. 


We will address here the problem of studying the higher secant
varieties σ_s(X), where X is one of the varieties above. We
have:

    σ_s(X) := ⋃ _P_1,…,P_s∈ X⟨ P_1,…,P_s⟩

i.e., σ_s(X) is the Zariski closure of the union of the ^s-1's, which are s-secant to X.

The problem of determining the dimensions of the higher secant
varieties of many classically-studied projective varieties (and also
projective varieties in general) is quite classical in algebraic geometry and has a long and interesting
history. By a simple count of parameters, the expected dimension of σ_s(X), for X⊂^N, is min{s( X)+(s-1),
N}. This is always an upper-bound of the actual dimension, and a variety X is said to be
defective, or s-defective, if there is a value s for which
the dimension of σ_s(X) is strictly smaller than the expected one;
the difference: 

    δ_s(X) := min{s( X)+(s-1), N} - σ_s(X)
 
is called the s-defectivity of X (or of σ_s(X)); a variety X for which some δ_s is positive is called defective.

The first interest in the secant variety σ_2(X) of a variety
X⊂^N lies in the fact that if σ_2(X) ≠
^N, then the projection of X from a generic point of ^N into
^N-1 is an isomorphism. This goes back to the XIX Century with the
discovery of a surface X ⊂ ^5, for which
σ_2(X) is a hypersurface, even though its expected dimension
is five. This is the Veronese surface, which is the only surface in ^5 with this property. The research on defective varieties has been quite a frequent
subject for classical algebraic geometers, e.g., see the works of
F. Palatini <cit.>, A. Terracini <cit.> and G.
 Scorza <cit.>.

It was then in the 1990s that two new articles marked a turning point
in the study about these questions and rekindled the interest in
these problems, namely the work of F. Zak and the one by J. Alexander and
A. Hirshowitz.

 Among many other things, like, e.g., proving Hartshorne's conjecture on linear normality, the outstanding paper of F. Zak <cit.> studied Severi varieties, i.e.,
non-linearly normal smooth n-dimensional subvarieties X⊂^N, with 2
3(N-1)=n. Zak found that all Severi varieties have defective σ_2(X), and, by
using invariant theory, classified all of them as follows.
 
Over an algebraically-closed field of characteristic zero, each Severi variety is
projectively equivalent to one of the following four projective
varieties:

	
  * Veronese surface ν_2(^2) ⊂^5;
	
  * Segre variety ν_1,1(^2 ×^2) ⊂^8;
	
  * Grassmann variety Gr(1,5)⊂^14;
	
  * Cartan variety E^16⊂^26.


Moreover, later in the paper, also Scorza varieties are classified,
which are maximal with respect to defectivity and which generalize
the result on Severi varieties.

The other significant work is the one done by J. Alexander and
A. Hirschowitz; see <cit.> and Theorem <ref> below. Although
not directly addressed to the study of secant varieties, they confirmed
the conjecture that, apart from the quadratic Veronese varieties and
a few well-known exceptions, all the Veronese varieties have
higher secant varieties of the expected dimension. In a sense, this
result completed a project that was underway for over 100 years (see
<cit.>).








 §.§ Secant Varieties and Tensor Decomposition


Tensors are multidimensional arrays of numbers and play an important role in numerous research areas including computational complexity, signal processing for telecommunications <cit.> and scientific data analysis <cit.>.
As specific examples, we can quote the complexity of matrix multiplication <cit.>, the P versus NP complexity problem 
 <cit.>, the study of entanglement in quantum physics <cit.>, matchgates in computer science <cit.>,
the study of phylogenetic invariants <cit.>, independent component analysis <cit.>, blind identification in signal processing <cit.>, branching structure in diffusion images <cit.> and other multilinear data analysis techniques in bioinformatics and spectroscopy <cit.>. Looking at this literature shows how knowledge about this subject used to 
be quite scattered and suffered a bit from
the fact that the same type of problem can be considered in different
areas using a different language.

In particular, tensor decomposition is nowadays an intensively-studied argument by many algebraic geometers and by more applied communities. Its main problem is the decomposition of a tensor with a given structure as a linear combination of decomposable tensors of the same structure called rank-one tensors. 














To be more precise:
let V_1, … , V_d be -vector spaces of dimensions n_1+1,…,n_d+1, respectively, and let
V = V_1^*⊗⋯⊗ V_d^* ≃ (V_1⊗…⊗ V_d)^*.
We call a decomposable, or rank-one, tensor an element of the type v^*_1⊗⋯⊗ v^*_d ∈. If T ∈V, one can ask: 

	What is the minimal length of an expression of T as a sum of decomposable tensors?


We call such an expression a tensor decomposition of T, and the answer to this question is usually referred to as the tensor rank of T. Note that, since V is a finite-dimensional vector space of dimension ∏ _i=1^d _ V_i,
which has a basis of decomposable tensors, it is quite trivial to see that every T ∈V can be written as the sum of finitely many decomposable tensors. Other natural questions to ask are:

What is the rank of a generic tensor in V? What is the dimension of the closure of the set of all tensors of tensor rank ≤ r?


Note that it is convenient to work up to scalar multiplication, i.e., in the projective space (), and the latter questions are indeed meant to be considered in the Zariski topology of (). This is the natural topology used in algebraic geometry, and it is defined such that closed subsets are zero loci of (homogeneous) polynomials and open subsets are always dense. In this terminology, an element of a family is said to be generic in that family if it lies in a proper Zariski open subset of the family. Hence, saying that a property holds for a generic tensor in (V) means that it holds on a proper Zariski subset of (V).


In the case d=2, tensors correspond to ordinary matrices, and the notion of tensor rank coincides with the usual one of the rank of matrices. Hence, the generic rank is the maximum one, and it is the same with respect to rows or to columns. When considering multidimensional tensors, we can check that in general, all these usual properties for tensor rank fail to hold; e.g., for
(2× 2× 2)-tensors, the generic tensor rank is two, but the
maximal one is three, and of course, it cannot be the dimension of the
space of “row vectors” in whatever direction.

 It is well known that studying the dimensions of the secant varieties to
Segre varieties gives a first idea of the stratification of V, or equivalently of (V),
 with respect to tensor rank. In fact, the Segre variety ν_1,…,1(^n_1×…×^n_d) can be seen as the projective variety in (), which parametrizes rank-one tensors, and consequently, the generic point of σ_s(ν_1,…,1(^n_1×⋯×^n_d)) parametrizes a tensor of tensor rank equal to s (e.g., see <cit.>).
 

If V_1 = ⋯ = V_d = V of dimension n+1, one can just consider symmetric or
skew-symmetric tensors. In the first case, we study the S^dV^*, which corresponds to the space of homogeneous polynomials in n+1 variables. Again, we have a notion of symmetric decomposable tensors, i.e, elements of the type (v^*)^d ∈ S^dV^*, which correspond to powers of linear forms. These are parametrized by the Veronese variety ν_d(^n)⊂(S^dV^*). In the skew-symmetric case, we consider ⋀^dV^*, whose skew-symmetric decomposable tensors are the elements of the form v^*_1∧…∧ v^*_d ∈⋀^dV^*. These are parametrized by the Grassmannian Gr(d,n+1) in its Plücker embedding. Hence, we get a notion of symmetric-rank and of ∧-rank for which one can ask the same questions as in the case of arbitrary tensors. Once again, these are translated into algebraic geometry problems on secant varieties of Veronese varieties and Grassmannians. 

Notice that actually, Veronese varieties 

embedded
in a projective space corresponding to (S^dV^*) 

can be thought of as sections of the Segre variety in 

 ((V^*)^⊗ d) defined by the (linear) equations given by the
symmetry relations.

Since the case of symmetric tensors is the one that has been classically considered more in depth, due to the fact that symmetric tensors correspond to homogeneous polynomials, we start from analyzing secant varieties of Veronese varieties in Section <ref>. Then, we pass to secant varieties of Segre varieties in Section <ref>. Then, Section <ref> is dedicated to varieties that parametrize other types of structured tensors, such as Grassmannians, which parametrize skew-symmetric tensors, Segre–Veronese varieties, which parametrize decomposable partially-symmetric tensors, Chow varieties, which parametrize homogeneous polynomials, which factorize as product of linear forms, varieties of powers, which parametrize homogeneous polynomials, which are pure k-th powers in the space of degree kd, or varieties that parametrize homogeneous polynomials with a certain prescribed factorization structure. In Section <ref>, we will consider other problems related to these kinds of questions, e.g., what is known about maximal ranks, how to find the actual value of (or bounds on) the rank of a given tensor, how to determine the number of minimal decompositions of a tensor, what is known about the equations of the secant varieties that we are considering or what kind of problems we meet when treating this problem over ℝ, a case that is of course very interesting for applications.




§ SYMMETRIC TENSORS AND VERONESE VARIETIES



A symmetric tensor T is an element of the space S^dV^*, where V^* is an (n+1)-dimensional -vector space and  is an algebraically-closed field.



















It is quite immediate to see that we can associate a degree d homogeneous polynomial in [x_0,…,x_n] with any symmetric tensor in S^dV^*. 

In this section, we address the problem of symmetric tensor decomposition.

 
 What is the smallest integer r such that a given symmetric tensor T ∈ S^dV^* can be written as a sum of r symmetric decomposable tensors, i.e., as a sum of r elements of the type (v^*)^⊗ d∈ (V^*)^⊗ d?


We call the answer to the latter question the symmetric rank of T. Equivalently, 

 
 What is the smallest integer r such that a given homogeneous polynomial F∈ S^dV^* (a (n+1)-ary d-ic
, in classical language) can be written as a sum of r d-th powers of linear forms?


We call the answer to the latter question the Waring rank, or simply rank, of F; denoted _sym(F). Whenever it will be relevant to recall the base field, it will be denoted by _ sym^(F). Since, as we have said, the space of symmetric tensors of a given format can be naturally seen as the space of homogeneous polynomials of a certain degree, we will use both names for the rank.

The name “Waring rank” comes from an old problem in number theory regarding expressions of integers as sums of powers; we will explain it in Section <ref>.


The first naive remark is that there are n+d  d coefficients a_i_0,⋯ ,i_n needed to write: 

    F=∑ a_i_0,⋯ ,i_nx_0^i_0⋯ x_n^i_n,
 
and r(n+1) coefficients b_i,j to write the same F as: 

    F=∑_i=1^r (b_i,0x_0+ ⋯ + b_i,nx_n)^d.
 
Therefore, for a general polynomial, the answer to the question should be that r has to be at least such that r(n+1)≥n+d  d. Then, the minimal value for which the previous inequality holds is ⌈1/n+1n+d d⌉. For n=2 and d=2, we know that this bound does not give the correct answer because a regular quadratic form in three variables cannot be written as a sum of two squares. On the other hand, a straightforward inspection shows that for binary cubics, i.e., d=3 and n=1, the generic rank is as expected. Therefore, the answer cannot be too simple. 


The most important general result on this problem has been obtained by J. Alexander and A. Hirschowitz, in 1995; see <cit.>. It says that the generic rank is as expected for forms of degree d≥ 3 in n≥ 1 variables except for a small number of peculiar pairs (n, d); see Theorem <ref>.


What about non-generic forms? As in the case of binary cubics, there are special forms that require a larger r, and these cases are still being investigated. Other presentations of this topic from different points of view can be found in <cit.>.


As anticipated in the Introduction, we introduce Veronese varieties, which parametrize homogeneous polynomials of symmetric-rank-one, i.e., powers of linear forms; see Section <ref>. Then, in order to study the symmetric-rank of a generic form, we will use the concept of secant varieties as defined in (<ref>). 

In fact, the order of the first secant that fills the ambient space will give the symmetric-rank of a generic form. The dimensions of secant varieties to Veronese varieties were completely classified by J. Alexander and A. Hirschowitz in <cit.> (Theorem <ref>). We will briefly review their proof since it provides a very important constructive method to compute dimensions of secant varieties that can be extended also to other kinds of varieties parameterizing different structured tensors. In order to do that, we need to introduce apolarity theory (Section <ref>) and the so-called Horace method (Sections <ref> and <ref>).

The second part of this section will be dedicated to a more algorithmic approach to these problems, and we will focus on the problem of computing the symmetric-rank of a given homogeneous polynomial. 

In the particular case of binary forms, there is a very well-known and classical result firstly obtained by J. J. Sylvester in the XIX Century. We will show a more modern reformulation of the same algorithm presented by G. Comas and M. Seiguer in <cit.> and a more efficient one presented in <cit.>; see Section <ref>. In Section <ref>, we will tackle the more general case of the computation of the symmetric-rank of any homogeneous polynomial, and we will show the only theoretical algorithm (to our knowledge) that is able to do so, which was developed by J. Brachat, P. Comon, B. Mourrain and E. Tsigaridas in <cit.> with its reformulation <cit.>. 

The last subsection of this section is dedicated to an overview of open problems.












 §.§ On Dimensions of Secant Varieties of Veronese Varieties



This section is entirely devoted to computing the symmetric-rank of a generic form, i.e., to the computation of the generic symmetric-rank. As anticipated, we approach the problem by computing dimensions of secant varieties of Veronese varieties. Recall that, in algebraic geometry, we say that a property holds for a generic form of degree d if it holds on a Zariski open, hence dense, subset of (S^dV^*).




  §.§.§ Waring problem for forms

The problem that we are presenting here takes its name from an old question in number theory. In 1770, E. Waring in <cit.> stated (without proofs) that:

“Every natural number can be written as sum of at most 9 positive cubes, 
Every natural number can be written as sum of at most 19 biquadratics.”


Moreover, he believed that: 

“For all integers d≥ 2, there exists a number g(d) such that each positive integer n∈ℤ^+ can be written as sum of the d-th powers of g(d) many positive integers, i.e., n=a_1^d+⋯ +a_g(d)^d with a_i≥ 0.”


E. Waring's belief was shown to be true by D. Hilbert in 1909, who proved that such a g(d) indeed exists for every d≥ 2. In fact, we know from the famous four-squares Lagrange theorem (1770) that g(2) = 4, and more recently, it has been proven that g(3) = 9 and g(4) = 19. However, the exact number for higher powers is not yet known in general. In <cit.>, H. Davenport proved that any sufficiently large integer can be written as a sum of 16 fourth powers. As a consequence, for any integer d ≥ 2, a new number G(d) has been defined, as the least number of d-th powers of positive integers to write any sufficiently large positive integer as their sum. Previously, C. F. Gauss proved that any integer congruent to seven modulo eight can be written as a sum of four squares, establishing that G(2) = g(2) = 4. Again, the exact value G(d) for higher powers is not known in general.


This fascinating problem of number theory was then formulated for homogeneous polynomials as follows.


Let  be an algebraically-closed field of characteristic zero.
We will work over the projective space ℙ^n=ℙV where V is an (n+1)-dimensional vector space over . 
We consider the polynomial ring S=[x_0,…,x_n] with the graded structure
S =⊕_d≥0S_d, where S_d=⟨ x_0^d,x_0^d-1x_1,…,x_n^d⟩ is the vector space of homogeneous polynomials, or forms, of degree d, which, as we said, can be also seen as the space S^dV of symmetric tensors of order d over V. 

In geometric language, those vector spaces S_d are called complete linear systems of
hypersurfaces of degree d in ℙ^n. Sometimes, we will write ℙS_d in order to mean the projectivization of S_d, namely ℙS_d will be a ℙ^n+d d-1 whose elements are classes of forms of degree d modulo scalar multiplication, i.e., [F]∈ℙS_d with F∈ S_d.

In analogy to the Waring problem for integer numbers, the so-called little Waring problem for forms is the following.
[Little Waring problem]
 Find the minimum s∈ℤ such that all forms F∈ S_d can be written as the sum of at most s d-th powers of linear forms.

The answer to the latter question is analogous to the number g(d) in the Waring problem for integers. At the same time, we can define an analogous number G(d), which considers decomposition in sums of powers of all numbers, but finitely many. In particular, the big Waring problem for forms can be formulated as follows.
[Big Waring problem]
Find the minimum s∈ℤ such that the generic form
F∈ S_d can be written as a sum of at most s d-th powers of linear
forms.

In order to know which elements of
S_d can be written as a sum of s d-th powers of linear forms,
we study the image of the map:

    ϕ_d,s : S_1×⋯× S_1_s⟶ S_d,   ϕ_d,s(L_1,…,L_s)=L_1^d+⋯+L_s^d.
 
In terms of maps ϕ_d,s, the little Waring problem (Problem <ref>) is to find the smallest s, such that Im(ϕ_d,s) = S_d. Analogously, to solve the big Waring problem (Problem <ref>), we require Im(ϕ_d,s) = S_d, which is equivalent to finding the minimal s such that (Im(ϕ_d,s)) =  S_d.

The map ϕ_d,s can be viewed as a polynomial map between affine spaces:

    ϕ_d,s : 𝔸^s(n+1)⟶𝔸^N,    with  N=n+d n.

In order to know the dimension of the image of such a map, we look
at its differential at a general point P of the domain:

    dϕ_d,s|_P:T_P(𝔸^s(n+1))⟶ T_ϕ_d,s(P)(𝔸^N).

Let P=(L_1,…,L_s)∈𝔸^s(n+1) and
v=(M_1,…,M_s) ∈ T_P(𝔸^s(n+1)) ≃𝔸^s(n+1), where L_i,M_i∈ S_1 for i=1,… ,s. Let us consider the
following parameterizations t ⟼
(L_1+M_1t,…,L_s+M_st) of a line 𝒞 passing through P whose tangent vector at P is M. The image of 𝒞 via ϕ_d,s is
ϕ_d,s
(L_1+M_1t,…,L_s+M_st)=∑_i=1^s
(L_i+M_it)^d. The tangent vector to ϕ_d,s(𝒞) in
ϕ_d,s(P) is: 

    .d/dt|_t=0( ∑_i=1^s
    (L_i+M_it)^d) = 
    ∑_i=1^s.d/dt|_t=0(L_i+M_it)^d = ∑_i=1^sdL_i^d-1M_i.

 Now, as
v=(M_1,…,M_s) varies in 𝔸^s(n+1), the tangent
vectors that we get span ⟨ L_1^d-1S_1,…,L_s^d-1S_1⟩. Therefore, we just proved the following.

 Let L_1,…,L_s be linear forms in S=[x_0,…,x_n], where L_i=a_i,0x_0+⋯+a_i,nx_n, and consider the map: 
    ϕ_d,s: S_1×⋯× S_1_s⟶ S_d,   ϕ_d,s (L_1,…,L_s)=L_1^d+⋯+L_s^d;

 then:
 
    rk(dϕ_d,s)|_(L_1,…,L_s)=_⟨ L_1^d-1S_1,…,L_s^d-1S_1⟩ .


It is very interesting to see how the problem of determining the latter dimension has been solved, because the solution involves many algebraic and geometric tools.





  §.§.§ Veronese Varieties



The first geometric objects that are related to our problem are the Veronese varieties.
We recall that a Veronese variety can be viewed as (is projectively equivalent to) the image of the following d-ple
 embedding of ^n, where all degree d monomials in n+1 variables appear in lexicographic order: 

    [                                       ν_d :                                         ℙ^n                                           ↪                                 ℙ^n+d  d -1;                                                                          [u_0: … : u_n]                                           ↦ [u_0^d: u_0^d-1u_1:
    u_0^d-1u_2: … : u_n^d].;                                             ]

With a slight abuse of notation, we can describe the Veronese map as follows:

    [            ν_d:  ℙ
    S_1= (ℙ^n)^*               ↪          ℙ^
    S_d =(ℙ^n+d d -1)^*;                             [L]               ↦           [L^d]                 ].

Let X_n,d := ν_d(^n) denote a Veronese variety.

Clearly, “ν_d as defined in (<ref>)” and “ν_d as defined in (<ref>)” are not the same map; indeed, from (<ref>),

    ν_d([L])    =ν_d([u_0x_0+ ⋯ + u_nx_n])= [L^d] = 
        = [u_0^d: du_0^d-1u_1:
    d 2u_0^d-1u_2: … : u_n^d]∈ S_d.

However, the two images are projectively equivalent. In order to see that, it is enough to consider the monomial basis of S_d given by: 

    {d α^α | α = (α_0,…,α_n) ∈^n+1, |α| = d}.


Given a set of variables x_0,…,x_n, we let ^α denote the monomial x_0^α_0⋯ x_n^α_n, for any α∈^n+1. Moreover, we write |α| = α_0+…+α_n for its degree. Furthermore, if |α| = d, we use the standard notation d α for the multi-nomial coefficient d!/α_0!⋯α_n!.

Therefore, we can view the Veronese variety either as the variety that
parametrizes d-th powers of linear forms or as the one parameterizing completely decomposable symmetric tensors.

[Twisted cubic] Let V=^2 and d=3, then:

    [                            ν_3:                             ℙ^1                               ↪                             ℙ^3;                                                       [a_0:a_1]                               ↦ [a_0^3:a_0^2a_1:a_0a_1^2:a_1^3] ].

If we take {z_0, … , z_3} to be homogeneous coordinates in ℙ^3, then the Veronese curve in ℙ^3 (classically known as twisted cubic) is given by the solutions of the following system of equations:

    {[  z_0z_2-z_1^2=0; z_0z_3-z_1z_2=0;  z_1z_3-z_2^2=0 ]. .

Observe that those equations can be obtained as the vanishing of all the maximal minors of the following matrix:

    ( [ z_0 z_1 z_2; z_1 z_2 z_3 ]).

Notice that the matrix (<ref>) can be obtained also as the defining matrix of the linear map:

    S^2V^* → S^1V,   ∂^2_x_i↦∂^2_x_i(F)

where F=∑_i=0^3 d  i^-1z_ix_0^3-ix_1^i and ∂_x_i:=∂/∂ x_i.

Another equivalent way to obtain (<ref>) is to use the so-called flattenings. We give here an intuitive idea about flattenings, which works only for this specific example.

 Write the 2× 2 × 2 tensor by putting in position ijk the variable z_i+j+k. This is an element of V^*⊗ V^* ⊗ V^*. There is an obvious isomorphism among the space of 2× 2 × 2 tensors V^*⊗ V^* ⊗ V^* and the space of 4× 2 matrices (V^* ⊗ V^*) ⊗ V^*. Intuitively, this can be done by slicing the 2× 2 × 2 tensor, keeping fixed the third index. This is one of the three obvious possible flattenings of a 2× 2 × 2 tensor: the other two flattenings are obtained by considering as fixed the first or the second index. Now, after having written all the possible three flattenings of the tensor, one could remove the redundant repeated columns and compute all maximal minors of the three matrices obtained by this process, and they will give the same ideal.


The phenomenon described in Example <ref> is a general fact. Indeed, Veronese varieties are always defined by 2× 2 minors of matrices constructed as (<ref>), which are usually called catalecticant matrices.
 
 Let F∈ S_d be a homogeneous polynomial of degree d in the polynomial ring S = [x_0,…,x_n]. For any i = 0,…,d, the (i,d-i)-th catalecticant matrix associated to F is the matrix representing the following linear maps in the standard monomial basis, i.e.,

    [ Cat_i,d-i(F) :          S_i^*              ⟶         S_d-i,;                        ∂^i_^α              ↦     ∂^i_^α(F),;                ]

where, for any α∈^n+1 with |α| = d-i, we denote ∂^d-i_^α := ∂^d-i/∂ x_0^α_0⋯∂ x_n^α_n.

Let {z_α | α∈^n+1,  |α| = d} be the set of coordinates on S^dV, where V is (n+1)-dimensional. The (i,d-i)-th catalecticant matrix of V is the n+i  n×n+d-i  n matrix whose rows are labeled by ℬ_i = {β∈^n+1 |  |β| = i} and columns are labeled by ℬ_d-i = {β∈^n+1 |  |β| = d-i}, given by:

    Cat_i,d-i(V) = 
    	[ z_β_1 + β_2 ]_β_1 ∈ℬ_i 
    β_2 ∈ℬ_d-i.



Clearly, the catalecticant matrix representing Cat_d-i,i(F) is the transpose of Cat_i,d-i(F). Moreover, the most possible square catalecticant matrix is Cat_⌊ d/2⌋ ,⌈ d/2⌉(F) (and its transpose).

 Let us describe briefly how to compute the ideal of any Veronese variety.

 A hypermatrix A=(a_i_1, … , i_d)_0≤ i_j≤ n,   j=1, … , d is said to be symmetric, or completely symmetric, if a_i_1, … , i_d=a_i_σ(1), … , i_σ(d) for all σ∈𝔖_d, where 𝔖_d is the permutation group of {1, … ,d}.


 Let H⊂ V^⊗ d be the n+d d-dimensional subspace of completely symmetric tensors of V^⊗ d, i.e., H is isomorphic to the symmetric algebra S^dV or the space of homogeneous polynomials of degree d in n+1 variables. 
Let S be a ring of coordinates of ℙ^ n+d d-1=ℙH obtained as the quotient S=S/I where S=[x_i_1, … , i_d]_0≤ i_j≤ n,   j=1, … ,d and I is the ideal generated by all: 
    x_i_1, … , i_d-x_i_σ(1), … , i_σ(d), ∀ σ∈𝔖_d.

The hypermatrix (x_i_1, … , i_d)_0≤ i_j≤ n,   j=1, … , d, whose entries are the generators of S, is said to be a generic symmetric hypermatrix.


Let A=(x_i_1, … , i_d)_0≤ i_j≤ n,   j=1, … , d be a generic symmetric hypermatrix, then it is a known result that the ideal of any Veronese variety is generated in degree two by the 2 × 2 minors of a generic symmetric hypermatrix, i.e., 

    I(ν_d(ℙ^n))=I_2(A) := (2 × 2  minors of  A) ⊂S̃.

See <cit.> for the set theoretical point of view. In <cit.>, the author proved that the ideal of the Veronese variety is generated by the two-minors of a particular catalecticant matrix. In his PhD thesis <cit.>, A. Parolin showed that the ideal generated by the two-minors of that catalecticant matrix is actually I_2(A), where A is a generic symmetric hypermatrix. 





  §.§.§ Secant Varieties

Now, we recall the basics on secant varieties.

 
Let X⊂ℙ^N be a projective variety
of dimension n. We define the s-th secant variety σ_s(X) of X as the closure of the union of all linear spaces spanned by s points lying on X, i.e.,

    σ_s(X):= ⋃_P_1,… ,P_s∈ X⟨ P_1,… ,P_s⟩⊂^N.

For any ⊂^n, ⟨⟩ denotes the linear span of , i.e., the smallest projective linear space containing .



The closure in the definition of secant varieties is necessary. Indeed, let L_1,L_2∈ S_1 be two homogeneous linear forms. The polynomial L_1^d-1L_2 is clearly in σ_2(ν_d(ℙ(V))) since we can write: 

    L_1^d-1L_2=lim_t→ 01/t((L_1+tL_2)^d-L_1^d);

however, if d>2, there are no M_1,M_2∈ S_1 such that L_1^d-1L_2=M_1^d+M_2^d. This computation represents a very standard concept of basic calculus: tangent lines are the limit of secant lines. Indeed, by (<ref>), the left-hand side of (<ref>) is a point on the tangent line to the Veronese variety at [L_1^d], while the elements inside the limit on the right-hand side of (<ref>) are lines secant to the Veronese variety at [L_1^d] and another moving point; see Figure <ref>.


From this definition, it is evident that the generic element of σ_s(X) is an element of some ⟨ P_1,… ,P_s⟩, with P_i ∈ X; hence, it is a linear combination of s elements of X. This is why secant varieties are used to model problems concerning additive decompositions, which motivates the following general definition.

	Let X⊂^N be a projective variety. For any P ∈^N, we define the X-rank of P as
	
    _X(P) = min{s  |  P ∈⟨ P_1,…,P_s ⟩,  for  P_1,…,P_s ∈ X},

	and we define the border X-rank of P as
	
    _X(P) = min{s  |  P ∈σ_s(X)}.
	

If X is a non-degenerate variety, i.e., it is not contained in a proper linear subspace of the ambient space, we obtain a chain of inclusions

    X ⊂σ_2(X) ⊂…⊂σ_s(X) = ^N.

 
The smallest s∈ℤ such that σ_s(X)=ℙ^N is called the generic X-rank. This is the X-rank of the generic point of the ambient space. 

The generic X-rank of X is an invariant of the embedded variety X.


As we described in (<ref>), the image of the d-uple Veronese embedding of ℙ^n =  S_1 can be viewed as the subvariety of S_d made by all forms, which can be written as d-th powers of linear forms. From this point of view, the generic rank s of the Veronese variety is the minimum integer such that the generic form of degree d in n+1 variables can be written as a sum of s powers of linear forms. In other words,

	the answer to the Big Waring problem (Problem <ref>) is the generic rank with respect to the d-uple Veronese embedding in S_d.


This is the reason why we want to study the problem of determining the dimension of
s-th secant varieties of an n-dimensional projective variety X⊂ℙ^N.


Let X^s:=X×⋯× X_s, X_0⊂ X be the open subset of regular points of X and:

    U_s(X) := { (P_1,… ,P_s)∈ X^s | ∀  i,  P_i∈ X_0,  and 
     the  P_i 's are independent}.

Therefore, for all (P_1,… ,P_s)∈ U_s(X), since the P_i's are linearly independent, the linear span
H=⟨ P_1,… ,P_s⟩ is a ℙ^ s-1. Consider the following incidence variety:
 
    ^s(X)={ (Q,H)∈ℙ^ N × U_s(X)   |   Q∈ H }.

If s≤ N+1, the dimension of that incidence variety is:

    (^s(X))=n(s-1)+n+s-1.

With this definition, we can consider the projection on the first factor:

    π_1:^s(X)→ℙ^ N;

 the s-th secant variety of X is just the closure of the image of this map, i.e.,

    σ_s(X)= Im(π_1:^s(X)→ℙ^ N).

Now, if (X)=n, it is clear that, while (^s(X))=ns+s-1,
the dimension of σ_s(X) can be smaller: it suffices that the
generic fiber of π_1 has positive dimension to impose
(σ_s(X))<n(s-1)+n+s-1. Therefore, it is a general fact that, if X⊂ℙ^ N and
(X)=n, then,

    (σ_s(X))≤min{N,sn+s-1 }.



A projective variety X⊂ℙ^ N of dimension n is said to
be s-defective if (σ_s(X))<min{ N,sn+s-1 }. If so, we call s-th defect of X the difference: 
    δ_s(X):= min{
    N,sn+s-1 } -(σ_s(X)).
 Moreover, if X is s-defective, then σ_s(X) is said to be defective. If σ_s(X) is not defective, i.e., δ_s(X)=0, then it is said to be regular or of expected dimension.


Alexander–Hirschowitz Theorem (<cit.>) tells us that the dimension of
the s-th secant varieties to Veronese varieties is not always the
expected one; moreover, they exhibit the list of all the defective cases.


Let X_n,d=ν_d( ℙ^n), for d≥ 2, be a Veronese variety. Then:
 
    (σ_s(X))=min{n+d  d  -1, sn+s-1}

 except for the following cases:
 
 
  (1) d=2, n≥ 2, s≤ n, where (σ_s(X))=min{n+2  2 -1, 2n+1-s  2};
 
  (2) d=3, n=4, s=7, where δ_s=1;
 
  (3) d=4, n=2, s=5, where δ_s=1;
 
  (4) d=4, n=3, s=9, where δ_s=1;
 
  (5) d=4, n=4, s=14, where δ_s=1.
 


Due to the importance of this theorem, we firstly give a historical review, then we will give the main steps of the idea of the proof. For this purpose, we will need to introduce many mathematical tools (apolarity in Section <ref> and fat points together with the Horace method in Section <ref>) and some other excursuses on a very interesting and famous conjecture (the so-called SHGH
 conjecture; see Conjecture <ref> and Conjecture <ref>) related to the techniques used in the proof of this theorem.



The following historical review can also be found in <cit.>. 

The quadric cases (d = 2) are classical. The first non-trivial exceptional case d=4 and n=2 was known already by Clebsch in 1860 <cit.>. He thought of the quartic as a quadric of quadrics and found that σ_5(ν_4(ℙ^2))⊊ℙ^14, whose dimension was not the expected one. Moreover, he found the condition that the elements of σ_5(ν_4(ℙ^2)) have to satisfy, i.e., he found the equation of the hypersurface σ_5(ν_4(ℙ^2)) ⊊ℙ^14: that condition was the vanishing of a 6× 6 determinant of a certain catalecticant matrix.

To our knowledge, the first list of all exceptional cases was described by Richmond in <cit.>, who showed all the defectivities, case by case, without finding any general method to describe all of them. It is remarkable that he could describe also the most difficult case of general quartics of ℙ^4. The same problem, but from a more geometric point of view, was at the same time studied and solved by Palatini in 1902–1903; see <cit.>. In particular, Palatini studied the general problem, proved the defectivity of the space of cubics in ℙ^4 and studied the case of n=2. He was also able to list all the defective cases.

The first work where the problem was treated in general is due to Campbell (in 1891; therefore, his work preceded those of Palatini, but in Palatini's papers, there is no evidence of knowledge of Campbell's work), who in <cit.>, found almost all the defective cases (except the last one) with very interesting, but not always correct arguments (the fact that the Campbell argument was wrong for n = 3 was claimed also in <cit.> in 1915).

 His approach is very close to the infinitesimal one of Terracini, who introduced in <cit.> a very simple and elegant argument (today known as Terracini's lemmas, the first of which will be displayed here as Lemma <ref>), which offered a completely new point of view in the field. Terracini showed again the case of n=2 in <cit.>. In <cit.>, he proved that the exceptional case of cubics in ℙ^4 can be solved by considering that the rational quartic through seven given points in ℙ^4 is the singular locus of its secant variety, which is a cubic hypersurface. In <cit.>, Terracini finally proved the case n=3 (in 2001, Roé, Zappalà and Baggio revised Terracini's argument, and they where able to present a rigorous proof for the case n = 3; see <cit.>). 

In 1931, Bronowski <cit.> tried to tackle the problem checking if a linear system has a vanishing Jacobian by a numerical criterion, but his argument was incomplete.

In 1985, Hirschowitz (<cit.>) proved again the cases n=2,3, and he introduced for the first time in the study of this problem the use of zero-dimensional schemes, which is the key point towards a complete solution of the problem (this will be the idea that we will follow in these notes). Alexander used this new and powerful idea of Hirschowitz, and in <cit.>, he proved the theorem for d ≥ 5.

Finally, in <cit.> (1992–1995), J. Alexander and A. Hirschowitz joined forces to complete the proof of Theorem <ref>. After this result, simplifications of the proof followed <cit.>.

 
After this historical excursus, we can now review the main steps of the proof of the Alexander–Hirschowitz theorem. As already mentioned, one of the main ingredients to prove is Terracini's lemma (see <cit.> or <cit.>), which gives an extremely powerful technique to compute the dimension of any secant variety.






  
Let X be an irreducible non-degenerate variety in ℙ^ N, and let P_1,… ,P_s be s generic points on X. Then, the tangent space to σ_s(X) at a generic point Q∈⟨ P_1,… ,P_s⟩ is the linear span in ℙ^ N of the tangent spaces T_P_i(X) to X at P_i, i=1,… ,s, i.e.,

    T_Q(σ_s(X)) = ⟨ T_P_1(X),… , T_P_s(X)⟩.



This “lemma” (we believe it is very reductive to call it a “lemma”) can be proven in many ways (for example, without any assumption on the characteristic of , or following Zak's book <cit.>). Here, we present a proof “made by hand”.

 We give this proof in the case of =ℂ, even though it works in general for any algebraically-closed field of characteristic zero. 

We have already used the notation X^s for X×⋯× X taken s times. Suppose that (X)=n. Let us consider the following incidence variety,

    ={ (P;P_1,… ,P_s) ∈ℙ^N × X^s  |  P ∈⟨ P_1,… ,P_s⟩
    
    P_1,… ,P_s∈ X}⊂ℙ^N× X^s,

and the two following projections,

    π_1:→σ_s(X) ,   π_2:→ X^s.
 
The dimension of X^s is clearly sn. If (P_1,… , P_s)∈ X^s, the fiber π_2^-1((P_1,… ,P_s)) is generically a ℙ^ s-1, s<N. Then, ()=sn+s-1. 

If the generic fiber of π_1 is finite, then σ_s(X) is regular. i.e., it has the expected dimension; otherwise, it is defective with a value of the defect that is exactly the dimension of the generic fiber.

Let (P_1,… , P_s)∈ X^s and suppose that each P_i∈ X ⊂ℙ^ N has coordinates P_i=[a_i,0:…:a_i,N], for i=1,… ,s. In an affine neighborhood U_i of P_i, for any i, the variety X can be locally parametrized with some rational functions f_i,j:^n+1→, with j=0,… ,N, that are zero at the origin. Hence, we write:

    X ⊃ U_i :  {[ x_0= a_i,0+ f_i,0(u_i,0,… ,u_i,n);                                 ⋮;  x_N=a_i,N+f_i,N(u_i,0, … ,u_i,n) ]. .

Now, we need a parametrization φ for σ_s(X). Consider the subspace spanned by s points of X, i.e.,

    ⟨ (a_1,0+f_1,0,… , a_1,N+f_1,N),… ,(a_s,0+f_s,0,… ,a_s,N+f_s,N)⟩,

where for simplicity of notation, we omit the dependence of the f_i,j on the variables u_i,j; thus, an element of this subspace is of the form: 

    λ_1(a_1,0+f_1,0,… , a_1,N+f_1,N)+⋯ +λ_s(a_s,0+f_s,0,… , a_s,N+f_s,N),

for some λ_1, … ,λ_s∈. We can assume λ_1=1. Therefore, a parametrization of the s-th secant variety to X in an affine neighborhood of the point P_1+λ_2P_2+…+λ_sP_s is given by: 

    (a_1,0+f_1,0,…,   a_1,N+f_1,N)+ 
        + (λ_2+t_2)(a_2,1-a_1,0+f_2,1-f_1,0,… ,a_2,N-a_1,N+f_2,N-f_1,N)+ 
        + ⋯ + 
        + (λ_s+t_s)(a_s,1-a_1,0+f_s,1-f_1,0,… ,a_s,N-a_1,N+f_s,N-f_1,N),

for some parameters t_2,… ,t_s. Therefore, in coordinates, the 
parametrization of σ_s(X) that we are looking for is the map φ: ^s(n+1)+s-1→^N+1 given by:
	

	(u_1,0,…,u_1,n,u_2,0,…,u_2,n,…,u_s,0,…,u_s,n,t_2,… ,t_s)
	
	
	
	(… , a_1,j+f_1,j+ (λ_2+t_2)(a_2,j-a_1,j+f_2,j-f_1,j)+⋯ +(λ_s+t_s)(a_s,j-a_1,j+f_s,j-f_1,j) ,…),
	

	where for simplicity, we have written only the j-th element of the image.
Therefore, we are able to write the Jacobian of φ. We are writing it in three blocks: the first one is (N+1)× (n+1); the second one is (N+1)× (s-1)(n+1); and the third one is (N+1)× (s-1):

    J_0(φ)=( [ (1-λ_2-⋯ -λ_s)∂ f_1,j/∂ u_1,k                             |            λ_i∂ f_i,j/∂ u_i,k                             |                  a_i,j- a_1,j ]),

with i=2,… ,s; j=0, … ,N and k=0,… ,n. Now, the first block is a basis of the (affine) tangent space to X at P_1, and in the second block, we can find the bases for the tangent spaces to X at P_2,… ,P_s; the rows of:

    ([ ∂ f_i,0/∂ u_i,0               ⋯ ∂ f_i,0/∂ a_i,N;               ⋮               ⋮; ∂ f_i,N/∂ a_i,0               ⋯ ∂ f_i,N/∂ a_i,N ])

give a basis for the (affine) tangent space of X at P_i.




The importance of Terracini's lemma to compute the dimension of any secant variety is extremely evident. One of the main ideas of Alexander and Hirshowitz in order to tackle the specific case of Veronese variety was to take advantage of the fact that Veronese varieties are embedded in the projective space of homogeneous polynomials. They firstly moved the problem from computing the dimension of a vector space (the tangent space to a secant variety) to the computation of the dimension of its dual (see Section <ref> for the precise notion of duality used in this context). Secondly, their punchline was to identify such a dual space with a certain degree part of a zero-dimensional scheme, whose Hilbert function can be computed by induction (almost always). We will be more clear on the whole technique in the sequel. Now we need to use the language of schemes.

 Schemes are locally-ringed spaces isomorphic to the spectrum of a commutative ring. Of course, this is not the right place to give a complete introduction to schemes. The reader interested in studying schemes can find the fundamental material in <cit.>. In any case, it is worth noting that we will always use only zero-dimensional schemes, i.e., “points”; therefore, for our purpose, it is sufficient to think of zero-dimensional schemes as points with a certain structure given by the vanishing of the polynomial equations appearing in the defining ideal. For example, a homogeneous ideal I contained in [x,y,z], which is defined by the forms vanishing on a degree d plane curve 𝒞 and on a tangent line to 𝒞 at one of its smooth points P, represents a zero-dimensional subscheme of the plane supported at P and of length two, since the degree of intersection among the curve and the tangent line is two at P (schemes of this kind are sometimes called jets).




 A fat point Z⊂ℙ^n is a zero-dimensional scheme, whose defining ideal is of the form ℘^m, where ℘ is the ideal of a simple point and m is a positive integer. In this case, we also say that Z is a m-fat point, and we usually denote it as mP. We call the scheme of fat points a union of fat points m_1P_1+⋯+m_sP_s, i.e., the zero-dimensional scheme defined by the ideal ℘_1^m_1∩⋯∩℘_s^m_s, where ℘_i is the prime ideal defining the point P_i, and the m_i's are positive integers.



	In the same notation as the latter definition, it is easy to show that F ∈℘^m if and only if ∂(F)(P) = 0, for any partial differential ∂ of order ≤ m-1. In other words, the hypersurfaces “vanishing” at the m-fat point mP are the hypersurfaces that are passing through P with multiplicity m, i.e., are singular at P of order m. 


 Let (X,ℒ) be an integral, polarized
scheme. If ℒ embeds X as a closed scheme in ℙ^ N,
then:

    (σ_s(X)) = N -  (h^0(ℐ_
    Z,X⊗ℒ)),
 where Z is the union of s
 generic two-fat
points in X.


 By Terracini's lemma, we have that, for generic points P_1,… ,P_s∈ X, 
    (σ_s(X)) = (⟨ T_P_1(X),… ,T_P_s(X)⟩).
 Since X is
embedded in ℙ^ (H^0(X,ℒ)^*) of dimension N, we can view the
elements of H^0(X,ℒ) as hyperplanes in ℙ^ N. The
hyperplanes that contain a space T_P_i(X) correspond to
elements in H^0(ℐ_ 2P_i,X⊗ℒ), since they
intersect X in a subscheme containing the first infinitesimal
neighborhood of P_i. Hence, the hyperplanes of ℙ^N containing
 ⟨ T_P_1(X),… ,T_P_s(X)⟩ are the sections of
H^0(ℐ_Z,X⊗ℒ), where Z is the scheme
union of the first infinitesimal neighborhoods in X of the
points P_i's. 


 A hyperplane H contains the tangent space to a non-degenerate projective variety X at a smooth point P if and only if the intersection X∩ H has a singular point at P. In fact, the tangent space T_P(X) to X at P has the same dimension of X and T_P(X∩ H)= H∩ T_P(X). Moreover, P is singular in H∩ X if and only if (T_P(X∩ H))≥ (X∩ H)= (X)-1, and this happens if and only if H ⊃ T_P(X).


[The Veronese surface of ^5 is defective] Consider the Veronese surface X_2,2 = ν_2(^2) in ℙ^ 5. We want to show that it is two-defective, with δ_2=1. In other words, since the expected dimension of σ_2(X_2,2) is 2· 2 + 1, i.e., we expect that σ_2(X_2,2) fills the ambient space, we want to prove that it is actually a hypersurface. This will imply that actually, it is not possible to write a generic ternary quadric as a sum of two squares, as expected by counting parameters, but at least three squares are necessary instead. 

Let P be a general point on the linear span ⟨ R,Q⟩ of two general points R,Q∈ X; hence, P ∈σ_2(X_2,2). By Terracini's lemma, T_P(σ_2(X_2,2))=⟨ T_R(X_2,2),T_Q(X_2,2)⟩. The expected dimension for σ_2(X_2,2) is five, so (T_P(σ_2(X_2,2)))<5 if and only if there exists a hyperplane H containing T_P(σ_2(X_2,2)). The previous remark tells us that this happens if and only if there exists a hyperplane H such that H∩ X_2,2 is singular at R, Q. Now, X_2,2 is the image of ℙ^2 via the map defined by the complete linear system of quadrics; hence, X_2,2∩ H is the image of a plane conic. Let R', Q' be the pre-images via ν_2 of R,Q respectively. Then, the double line defined by R',Q' is a conic, which is singular at R',Q'. Since the double line ⟨ R',Q'⟩ is the only plane conic that is singular at R',Q', we can say that (T_P(σ_2(X_2,2)))=4<5; hence, σ_2(X_2,2) is defective with defect equal to zero. 

Since the two-Veronese surface is defined by the complete linear system of quadrics, Corollary <ref> allows us to rephrase the defectivity of σ_2(X_2,2) in terms of the number of conditions imposed by two-fat points to forms of degree two; i.e., we say that


two two-fat points of ℙ^2 do not impose independent conditions on ternary quadrics.


As we have recalled above, imposing the vanishing at the two-fat point means to impose the annihilation of all partial derivatives of first order. In ℙ^2, these are three linear conditions on the space of quadrics. Since we are considering a scheme of two two-fat points, we have six linear conditions to impose on the six-dimensional linear space of ternary quadrics; in this sense, we expect to have no plane cubic passing through two two-fat points. However, since the double line is a conic passing doubly thorough the two two-fat points, we have that the six linear conditions are not independent. We will come back in the next sections on this relation between the conditions imposed by a scheme of fat points and the defectiveness of secant varieties.


Corollary <ref> can be generalized to non-complete linear systems on X.

 Let D be any divisor of an irreducible projective variety X. With |D|, we indicate the complete linear system defined by D. Let V⊂ |D| be a linear system. We use the notation:

    V(m_1P_1, … , m_sP_s)

for the subsystem of divisors of V passing through the fixed points P_1, … , P_s with multiplicities at least m_1, … , m_s respectively.


When the multiplicities m_i are equal to two, for i=1,… ,s, since a two-fat point in ^n gives n+1 linear conditions, in general, we expect that, if (X)=n, then:

    (V(2P_1, … , 2P_s))=  (V)- s(n+1).

Suppose that V is associated with a morphism φ_V:X_0→ℙ^ r (if (V)=r), which is an embedding on a dense open set X_0⊂ X. We will consider the variety φ_V(X_0).

The problem of computing (V(2P_1, … , 2P_s)) is equivalent to that one of computing the dimension of the s-th secant variety to φ_V(X_0).

 Let X be an integral scheme and V be a linear system on X such that the rational function φ_V:Xℙ^ r associated with V is an embedding on a dense open subset X_0 of X
. Then, σ_s(φ_V(X_0)) is defective if and only if for general points, we have P_1, … , P_s∈ X:

    (V(2P_1, … , 2P_s))>min{-1, r-s(n+1)} .
 

This statement can be reformulated via apolarity, as we will see in the next section.




  §.§.§ Apolarity


This section is an exposition of inverse systems techniques, and it follows <cit.>.

As already anticipated at the end of the proof of Terracini's lemma, the whole Alexander and Hirshowitz technique to compute the dimensions of secant varieties of Veronese varieties is based on the computation of the dual space to the tangent space to σ_s(ν_d(ℙ^n)) at a generic point. Such a duality is the apolarity action that we are going to define. 

[Apolarity action]
Let S=[x_0,…,x_n] and
R=[y_0,…,y_n] be polynomial rings and consider the action of R_1 on S_1 and of R_1 on S_1 defined by:

    y_i∘ x_j= (∂/∂ x_i)
     (x_j)={[       0, if  i≠ j;       1,  if  i=j ]. ;

i.e., we view the polynomials of R_1 as “partial derivative operators” on S_1.
 
 
Now, we extend this action to the whole rings R and S by linearity and using properties
of differentiation. Hence, we get the apolarity action:

    ∘: R_i× S_j⟶ S_j-i

where:

    ^α∘^β=
    ∏_i=1^n (β_i)!/(β_i-α_i)!^β - α,     if α≤β; 
    
      0,    otherwise.
 
for α, β∈ℕ^n+1, α = (α_0,...,α_n), β = (β_0,…,β_n), where we use the notation α≤β if and only if a_i≤ b_i for all
i=0,… ,n, which is equivalent to the condition that x^α divides x^β in S.
 Here, are some basic remarks on apolarity action:


  * the apolar action of R on S makes S a (non-finitely generated) R-module (but the converse is not true);
 
  * the apolar action of R on S lowers the degree; in particular, given F ∈ S_d, the (i,d-i)-th catalecticant matrix (see Definition <ref>) is the matrix of the following linear map induced by the apolar action
   
    Cat_i,d-i(F) : R_i ⟶ S_d-i,    G ↦ G ∘ F;

 
  * the apolarity action induces a non-singular -bilinear pairing:
 
    R_j× S_j⟶,   ∀  j ∈,

that induces two bilinear maps (Let V × W ⟶ be a -bilinear parity given by
 v × w ⟶ v ∘ w. It induces two -bilinear maps:
 (1) ϕ : V ⟶ Hom_(W,) such that
  ϕ (v) := ϕ_v and ϕ_v(w)=v ∘ w and χ :
  W ⟶ Hom_(V,) such that χ (w) :
  = χ_w and χ_w(v)=v ∘ w; (2) V × W ⟶
  is not singular iff for all the bases
   {w_1,…,w_n} of W, the matrix (b_ij=v_i∘ w_j) is invertible.).



 Let I be a homogeneous ideal of R. The inverse system I^-1 of I
 is the R-submodule of S containing all the elements of S, which are annihilated by I via the apolarity action.


Here are some basic remarks about inverse systems:


  * if I=(G_1,…,G_t)⊂ R and F ∈ S, then:
 
    F ∈ I^-1⟺ G_1∘ F = ⋯ = G_t∘
     F = 0,
 finding all such F's amounts to finding all the polynomial solutions for the differential equations defined by the G_i's, so one can notice that determining I^-1 is equivalent to solving (with
polynomial solutions) a finite set of differential equations;

  * I^-1 is a graded submodule of S, but it is not
necessarily multiplicatively closed; hence in general, I^-1 is not
an ideal of S. 



Now, we need to recall a few facts on Hilbert functions and Hilbert series.


Let X ⊂ℙ^n be a closed subscheme whose
defining homogeneous ideal is I:=I(X) ⊂ S=[x_0,…,x_n]. Let A= S
/I be the homogeneous coordinate ring of X, and A_d will be its degree d component.


The Hilbert function of the scheme X is the numeric function:

    (X,·):ℕ→ℕ;


    (X,d)= _(A_d) = _(S_d)-_(I_d).

The Hilbert series of X is the generating power series:

    (X;z) = ∑_d ∈(X,d) t^d ∈[[z]].



In the following, the importance of inverse systems for a particular choice of the ideal I will be given by the following result.

 The dimension of the part of degree d of the
inverse system of an ideal I⊂ R is the Hilbert function of
R/I in degree d:

    _(I^-1)_d=codim_(I_d)=(R/I,d).




If V × W → is a non-degenerate bilinear form and 
U is a subspace of V, then with U^, we denote the subspace of
W given by: 
    U^= {w ∈ W   |   v ∘ w =0  ∀  v ∈ U }.
 With this definition, we observe that:
 

  * if we consider the bilinear map in (<ref>) and an ideal I⊂ R, then:
 
(I^-1)_d≅ I_d^.

  * moreover, if I is a monomial ideal, then:

I_d^=⟨monomials of R_d that are not in
I_d⟩;

  * for any two ideals I,J⊂ R: (I ∩ J)^-1=I^-1+J^-1.


If I=℘_1^m_1+1∩⋯∩℘_s^m_s+1⊂
R = [y_0,…,y_n] is the defining ideal of the scheme of fat points m_1P_1+⋯+m_sP_s ∈^n, where P_i=[p_i_0:p_i_1:…:p_in]∈^n, and if
L_P_i=p_i_0x_0+p_i_1x_1+⋯+p_i_nx_n∈
S=[x_0,…,x_n], then:
  
    (I^-1)_d= 
      
      S_d,     for  d ≤max{m_i}, 
    
     L_P_1^d-m_1S_m_1+⋯+L_P_s^d-m_sS_m_s,     for  d ≥max{m_i+1},

and also: 

    (R/I,d)     =_(I^-1)_d = 
        = _ S_d,     for  d ≤max{m_i}
    _⟨ L_P_1^d-m_1S_m_1,…,L_P_s^d-m_sS_m_s⟩,     for  d ≥max{m_i+1}

This last result gives the following link between the Hilbert function of a
set of fat points and ideals generated by sums of powers of linear
forms.


Let I=℘_1^m_1+1∩⋯∩℘_s^m_s+1⊂ R = [y_0,… ,y_n], then the inverse system (I^-1)_d⊂ S_d=[x_0,… ,x_n]_d is the d-th graded part
of the ideal (L_P_1^d-m_1,…,L_P_s^d-m_s)⊂ S,
for d≥max{ m_i+1 ,   i=1,… ,s}.
 


Finally, the link between the big Waring problem (Problem <ref>) and inverse systems is clear. If in (<ref>), all the m_i's are equal to one, the dimension of the vector space ⟨ L_P_1^d-1S_1,… ,L_P_s^d-1S_1⟩ is at the same time the Hilbert function of the inverse system of a scheme of s double fat points and the rank of the differential of the application ϕ defined in (<ref>). 


 Let L_1,…,L_s be linear forms of S = [x_0,…,x_n] such that:
 
    L_i=a_i_0x_0+⋯+a_i_nx_n,

and let P_1,…,P_s∈ℙ^ n such that
 P_i=[a_i_0,…,a_i_n].
Let ℘_i⊂ R = [y_0,…,y_n] be the prime ideal
associated with P_i, for i=1,…,s, and let:
 
    ϕ_s,d: S_1×⋯× S_1_s⟶ S_d

 with
 ϕ_s,d (L_1,…,L_s)=L_1^d+⋯+L_s^d. Then,
 
    (dϕ_s,d)|_(L_1,…,L_s)=_⟨ L_1^d-1S_1,…,L_s^d-1S_1⟩.
 Moreover, by (<ref>), we have:
 
    _ (⟨ L_1^d-1S_1,…,L_s^d-1S_1⟩)=( R/℘_1^2∩⋯∩℘_s^2,d ).



Now, it is quite easy to see that: 

    ⟨ T_P_1X_n,d, … ,T_P_sX_n,d⟩=⟨ L_1^d-1S_1,…,L_s^d-1S_1⟩.

Therefore, putting together Terracini's lemma and Proposition <ref>, if we assume the L_i's (hence, the P_i's) to be generic, we get:

    (σ_s(   X_n,d)) = ⟨ T_P_1X_n,d, … ,T_P_sX_n,d⟩ = 
        = _⟨ L_1^d-1S_1,…,L_s^d-1S_1⟩-1 = (R/(℘_1^2 ∩⋯∩℘_s^2),d)-1.


 Let P∈ℙ^ n, ℘⊂ S be its representative prime ideal and f∈ S. Then, the order of all partial derivatives of f vanishing in P is almost t if and only if
 f ∈℘^t+1, i.e., P is a singular point of V(f) of multiplicity greater than or equal to t+1. Therefore,
 
    (S/ ℘^t,d)=
     d+n  n    if  d < t; 
    t-1+n  n    if  d ≥ t.

It is easy to conclude that one t-fat point of ℙ^ n has the same Hilbert function of
t-1+n n generic distinct points of ℙ^ n. Therefore, (X_n,d)=(S/ ℘^2,d)-1=(n+1)-1. This reflects the fact that Veronese varieties are never one-defective, or, equivalently, since X_n,d = σ_1(X_n,d), that Veronese varieties are never defective: they always have the expected dimension 1· n+1-1.


Let P_1,P_2 be two points of ℙ^2, ℘_i⊂ S=[x_0,x_1,x_2] their associated prime ideals and m_1=m_2=2, so that I= ℘_1^2∩℘_2^2. Is the Hilbert function of I equal to the Hilbert function of six points of ℙ^ 2 in general position? No; indeed, the Hilbert series of six general points of ℙ^ 2 is 1+3z+6∑_i ≥ 2z^i. This means that I should not contain conics, but this is clearly false because the double line through P_1 and P_2 is contained in I. By (<ref>), this implies that σ_2(ν_2(ℙ^2)) ⊂^5 is defective, i.e., it is a hypersurface, while it was expected to fill all the ambient space.

 
[Fröberg–Iarrobino's conjecture]
	Ideals generated by powers of linear forms are usually called power ideals. Besides the connection with fat points and secant varieties, they are related to several areas of algebra, geometry and combinatorics; see <cit.>. Of particular interest is their Hilbert function and Hilbert series. In <cit.>, Fröberg gave a lexicographic inequality for the Hilbert series of homogeneous ideals in terms of their number of variables, number of generators and their degrees. That is, if I = (G_1,…,G_s) ⊂ S=[x_0,…,x_n] with (G_i) = d_i, for i = 1,…,s,
	
    (S/I;z) ≽_ Lex⌈∏_i=1^s (1-z^d_i)/(1-z)^n+1⌉,

	where ⌈·⌉ denotes the truncation of the power series at the first non-positive term. Fröberg conjectured that equality holds generically, i.e., it holds on a non-empty Zariski open subset of S_d_1×…× S_d_s. By semicontinuity, fixing all the numeric parameters (n;d_1,…,d_s), it is enough to exhibit one ideal for which the equality holds in order to prove the conjecture for those parameters. In <cit.> (Main Conjecture 0.6), Iarrobino suggested to look to power ideals and asserted that, except for a list of cases, their Hilbert series coincides with the right-hand-side of (<ref>). By (<ref>), such a conjecture can be translated as a conjecture on the Hilbert function of schemes of fat points. This is usually referred to as the Fröberg–Iarrobino conjecture; for a detailed exposition on this geometric interpretation of Fröberg and Iarrobino's conjectures, we refer to <cit.>. As we will see in the next section, computing the Hilbert series of schemes of fat points is a very difficult and largely open problem.

 
Back to our problem of giving the outline of the proof of Alexander and Hirshowitz Theorem (Theorem <ref>): Proposition <ref> clearly shows that the computation of T_Q(σ_s(ν_d(ℙ^n))) relies on the knowledge of the Hilbert function of schemes of double fat points. Computing the Hilbert function of fat points is in general a very hard problem. In ℙ^2, there is an extremely interesting and still open conjecture (the SHGH conjecture). The interplay with such a conjecture with the secant varieties is strong, and we deserve to spend a few words on that conjecture and related aspects.
 
 


 §.§ Fat Points in the Plane and SHGH Conjecture

The general problem of determining if a set of generic points P_1,… ,P_s in the plane, each with a structure of m_i-fat point, has the expected Hilbert function is still an open one. There is only a conjecture due first to B. Segre in 1961 <cit.>, then rephrased by B. Harbourne in 1986 <cit.>, A. Gimigliano in 1987 <cit.>, A. Hirschowitz in 1989 <cit.> and others. It describes how the elements of a sublinear system of a linear system ℒ formed by all divisors in ℒ having multiplicity at least m_i at the points P_1, … , P_s, look when the linear system does not have the expected dimension, i.e., the sublinear system depends on fewer parameters than expected. For the sake of completeness, we present the different formulations of the same conjecture, but the fact that they are all equivalent is not a trivial fact; see <cit.>. 

Our brief presentation is taken from <cit.>, which we suggest as excellent and very instructive deepening on this topic.


Let X be a smooth, irreducible, projective, complex variety of dimension n. Let ℒ be a complete linear system of divisors on X. Fix P_1,… ,P_s distinct points on X and m_1,… ,m_s positive integers. We denote by ℒ(-∑_i=1^s m_iP_i) the sublinear system of ℒ formed by all divisors in ℒ having multiplicity at least m_i at P_i, i = 1,… ,s. Since a point of multiplicity m imposes m+n-1  n conditions on the divisors of ℒ, it makes sense to define the expected dimension of ℒ(-∑_i=1^s m_iP_i) as:

    (ℒ(-∑_i=1^s m_iP_i)):=max{ (ℒ)- ∑_i=1^sm_i+n-1  n,-1}.

 If ℒ(-∑_i=1^s m_iP_i) is a linear system whose dimension is not the expected one, it is said to be a special linear system.
Classifying special systems is equivalent to determining the Hilbert function of the zero-dimensional subscheme of ℙ^n given s general fat points of given multiplicities.

A first reduction of this problem is to consider particular varieties X and linear systems ℒ on them. From this point of view, the first obvious choice is to take X = ℙ^n and ℒ = ℒ_n,d := |𝒪_ℙ^n (d)|, the system of all hypersurfaces of degree d in ℙ^n. In this language, ℒ_n,d(-∑_i=1^s m_iP_i) are the hypersurfaces of degree d in n+1 variables passing through P_1, … , P_s with multiplicities m_1, … , m_s, respectively. 


The SHGH conjecture describes how the elements of ℒ_2,d(-∑_i=1^s m_iP_i) look when not having the expected dimension; here are two formulations of this. 


If ℒ_2,d(-∑_i=1^s m_iP_i) is a special linear system, then there is a fixed double component for all curves through the scheme of fat points defined by ℘_1^m_1∩⋯∩℘_s^m_s.



Consider ℒ_2,d(-∑_i=1^s m_iP_i). Then, one has the following possibilities:


  * the system is non-special, and its general member is irreducible; 

  * the system is non-special; its general member is non-reduced, reducible; 
its fixed components are all rational curves, except for at most one (this 
may occur only if the system is zero-dimensional); and the general member 
of its movable part is either irreducible or composed of rational curves in 
a pencil; 

  * the system is non-special of dimension zero and consists of a unique multiple 
elliptic curve; 

  * the system is special, and it has some multiple rational curve as a fixed 
component. 



This problem is related to the question of what self-intersections occur
for reduced irreducible curves
on the surface X_s obtained by blowing up the projective plane at the s points.
Blowing up the points introduces rational curves (infinitely many 
when s>8) of self-intersection -1.
Each curve ⊂ X_s corresponds to a curve D_⊂ℙ^2 of some degree d
vanishing to orders m_i at the s points:

    ℙ^2  X_s,    D_↦,

and the self-intersection ^2 is d^2-m_1^2-⋯-m_s^2 if D_∈ℒ_2,d(-∑_i=1^s m_iP_i). 


 An example of a curve D_ corresponding to a curve  such that ^2=-1 on X_s
is the line through two of the points; in this case,
d=1, m_1=m_2=1 and m_i=0 for i>2, so we have 
d^2-m_1^2-⋯-m_s^2=-1.

According to the SHGH conjecture, these (-1)-curves should be the 
only reduced irreducible curves of negative self-intersection,
but proving that there are no others turns out to be itself very hard
and is still open.

 
Let P_1,… ,P_s be s points of ℙ^ n in general position.
The expected dimension of ℒ(-∑_i=1^s m_iP_i) is: 

    (ℒ(-∑_i=1^s m_iP_i)):=max{(ℒ(-∑_i=1^s m_iP_i) ),-1},

where:

    (ℒ(-∑_i=1^s m_iP_i) ):=n+d d-1-∑_i=1^sm_i+n-1 n,

is the virtual dimension of ℒ(-∑_i=1^s m_iP_i). 

Consider the blow-up π: ℙ^2 ℙ^2 of the plane at the points P_1, … , P_s. Let E_1,… ,E_s be the exceptional divisors corresponding to the blown-up points P_1, … , P_s, and let H be the pull-back of a general line of ℙ^2 via π. The strict transform of the system ℒ := ℒ_2,d(∑_i=1^sm_iP_i) is the system ℒ= |dH - ∑_i=1^sm_iE_i|. 
Consider two linear systems ℒ := ℒ_2,d(∑_i=1^sm_iP_i) and ℒ' := ℒ_2,d(∑_i=1^sm'_iP_i). Their intersection product is defined by using the intersection product of their strict transforms on ℙ^2, i.e.,

    ℒ·ℒ'=ℒ·ℒ'=dd'-∑_i=1^sm_im'_i.

Furthermore, consider the anticanonical
class -K := -K_ℙ^2 of ℙ^2 corresponding to the linear system ℒ_2,d(-∑^s_i=1 P_i), which, by abusing notation, we also denote by -K.
The adjunction formula tells us that the arithmetic genus p_a(ℒ) of a curve
in ℒ is:

    p_a(ℒ)=ℒ·(ℒ+K)/2+1=d-1  2∑_i=1^sm_i  2,

which one defines to be the geometric genus of ℒ, denoted g_ℒ.


This is the classical Clebsch formula. Then, Riemann–Roch says that:

    (ℒ) = (ℒ)    = ℒ·(ℒ-K) +h^1(ℙ^2,L)-h^2(ℙ^2,L) =
        = ℒ^2 - g_ℒ + 1 + h^1(ℙ^2,ℒ) = (ℒ) + h^1(ℙ^2,ℒ)

because clearly, h^2(ℙ^2, ℒ) = 0. Hence,

    ℒ is non-special if and only if  h^0 (ℙ^2 , ℒ) · h^1 (ℙ^2 , ℒ) = 0 .


Now, we can see how, in this setting, special systems can naturally arise. Let us look for an irreducible curve  on ℙ^2, corresponding to a linear system ℒ on ℙ^2, which is expected to exist, but, for example, its double is not expected to exist. It translates into the following set of inequalities:

    (ℒ)≥ 0;
    
    g_ℒ≥ 0;
    (2ℒ)≤ -1;

which is equivalent to:

    ^2 - · K≥ 0;
    ^2 + · K≥ -2;
    
    2^2 -· K ≤ 0;
    .

and it has the only solution:

    ^2 = · K = -1,

which makes all the above inequalities equalities. Accordingly,  is a rational curve, i.e., a curve of genus zero, with self-intersection -1, i.e., a (-1)-curve. A famous theorem of Castelnuovo's (see <cit.> (p. 27)) says that these are the only curves that can be contracted to smooth points via a birational morphism of the surface on which they lie to another surface. By abusing terminology, the curve Γ⊂ℙ^2 corresponding to  is also called a (-1)-curve.

More generally, one has special linear systems in the following situation. Let ℒ be a linear system on ℙ^2, which is not empty; let  be a (-1)-curve on ℙ^2 corresponding to a curve Γ on ℙ^2, such that ℒ· = -N < 0. Then,  (respectively, Γ) splits off with multiplicity N as a fixed component from all curves of ℒ (respectively, ℒ), and one has:

    ℒ=N+ℳ,    (respectively, ℒ=NΓ+ℳ ),

where ℳ (respectively, ℳ) is the residual linear system. Then, one computes:

    (ℒ) = (ℳ) ≥(ℳ) = (ℒ) + N 2,

and therefore, if N≥ 2, then ℒ is special.

 
One immediately finds examples of special systems of this type by starting from the (-1)-curves of the previous example. For instance, consider ℒ := ℒ_2,2d(- ∑_i=1^5 dP_i), which is not empty, consisting of the conic ℒ_2,2(∑^d_i=1 P_i) counted d times, though it has virtual dimension -d 2.


Even more generally, consider a linear system ℒ on ℙ^2, which is not empty, _1,… ,_k some (-1)-curves on 
ℙ^2 corresponding to curves Γ_1,… ,Γ_k on ℙ^2, such that ℒ·_i = -N_i < 0, i = 1,… ,k. Then, for i = 1,…,k,

    ℒ=∑_i=1^k N_i Γ_i + ℳ,   ℒ =
    ∑^k_i=1
    N_i _i + ℳ,    and   ℳ·_i =0.
 

As before, ℒ is special as soon as there is an i = 1,… ,k such that N_i ≥ 2. Furthermore, _i ·_j =δ_i,j, because the union of two meeting (-1)-curves moves, according to the Riemann–Roch theorem, in a linear system of positive dimension on ℙ^2, and therefore, it cannot be fixed for ℒ. In this situation, the reducible curve := ∑^k_i=1_i (respectively, Γ := ∑^k_i=1 N_iΓ_i) is called a (-1)-configuration on ℙ^2 (respectively, on ℙ^2).


Consider ℒ := ℒ_2,d(-m_0P_0 -∑^ s_i=1 m_iP_i), with m_0 + m_i = d + N_i,
N_i ≥ 1. Let Γ_i be the line joining P_0, P_i. It splits off N_i times from ℒ. Hence: 
    ℒ = ∑^s_i=1 N_iΓ_i + ℒ_2,d -∑_i=1^s N_i(-(m_0 -∑^s_i=1 N_i)P_0 -∑^s_i=1(m_i -N_i)P_i).
 If we require the latter system to have non-negative virtual dimension, e.g., d≥∑_i=1^sm_i, if m_0 = d and some N_i > 1, we have as many special systems as we want.



A linear system ℒ on ℙ^2 is (-1)-reducible if ℒ = ∑^ k_i=1 N_i_i +ℳ, where = ∑^ k_i=1_i is a (-1)-configuration, ℳ·_i = 0, for all i = 1,… , k and (ℳ) ≥ 0.
The system ℒ is called (-1)-special if, in addition, there is an i = 1,… , k such that N_i > 1.


 A linear system of plane curves ℒ_2,d(-∑_i=1^s m_iP_i) with general multiple base points is special if and only if it is (-1)-special, i.e., it contains some multiple rational curve of self-intersection -1 in the base locus. 


No special system has been discovered except (-1)-special systems.

Eventually, we signal a concise version of the conjecture (see <cit.> (Conjecture 3.3)), which involves only a numerical condition. 

 
A linear system of plane curves ℒ_2,d(-∑_i=1^s m_iP_i) with general multiple base points and such that m_1≥ m_2≥…≥ m_s ≥ 0 and d ≥ m_1+m_2+m_3 is always non-special.



The idea of this conjecture comes from Conjecture <ref> and by working on the surface X =^̃2̃, which is the blow up of ^2 at the points P_i; in this way, the linear system ℒ_2,d(-∑_i=1^s m_iP_i) corresponds to the linear system ℒ̃ = dE_0 - m_1E_1 - …-E_s on X, where (E_0,E_1,… , E_s) is a basis for Pic(X), and E_0 is the strict transform of a generic line of ^2, while the divisors E_1,…, E_s are the exceptional divisors on P_1,…, P_s. If we assume that the only special linear systems ℒ_2,d(-∑_i=1^s m_iP_i) are those that contain a fixed multiple (-1)-curve, this would be the same for ℒ̃ in Pic(X), but this implies that either we have m_s < -1, or we can apply Cremona transforms until the fixed multiple (-1)-curve becomes of type -m'_iE'_i in Pic(X), where the E'_i's are exceptional divisors in a new basis for Pic(X). Our conditions in Conjecture <ref> prevent these possibilities, since the m_i are positive and the condition d ≥ m_1+m_2+m_3 implies that, by applying a Cremona transform, the degree of a divisor with respect to the new basis cannot decrease (it goes from d to d'=2d-m_i-m_j-m_k, if the Cremona transform is based on P_i, P_j and P_k), hence cannot become of degree zero (as -m'_iE'_i would be).


One could hope to address a weaker version of this problem. Nagata, in connection with his 
negative solution of the fourteenth Hilbert problem, made such a conjecture, 

 The linear system ℒ_2,d(-∑_i=1^s m_iP_i) is empty as soon as s≥ 10 and d≤√(s).


Conjecture <ref> is weaker 
than Conjecture <ref>, yet still open for every non-square n≥ 10. Nagata's conjecture does not rule out the occurrence of 
curves of self-intersection less than -1, but it does rule out the worst of them.
In particular, Nagata's conjecture asserts that d^2≥ sm^2 must hold
when s≥10, where m=(m_1+⋯+m_s)/s. Thus, perhaps there are curves
with d^2-m_1^2-⋯-m_s^2<0, such as the (-1)-curves mentioned above,
but d^2-m_1^2-⋯-m_s^2 is (conjecturally) only as negative 
as is allowed by the condition that after averaging the multiplicities m_i 
for n≥ 10, one must have d^2-sm^2≥ 0. 



Now, we want to find a method to study the Hilbert function of a zero-dimensional scheme. One of the most classical methods is the so-called Horace method (<cit.>), which has also been extended with the Horace differential technique and led J. Alexander and A. Hirschowitz to prove Theorem <ref>. We explain these methods in Sections <ref> and <ref>, respectively, and we resume in Section <ref> the main steps of the Alexander–Hirschowitz theorem.





  §.§.§ La Méthode D'Horace


In this section, we present the so-called Horace method. 
It takes this name from the ancient Roman legend (and a play by Corneille: Horace, 1639) about the duel between three Roman brothers, the “Orazi”, and three brothers from the enemy town of Albalonga, the “Curiazi”. The winners were to have their town take over the other one. After the first clash among them, two of the Orazi died, while the third remained alive and unscathed, while the Curiazi were all wounded, the first one slightly, the second more severely and the third quite badly. There was no way that the survivor of the Orazi could beat the other three, even if they were injured, but the Roman took to his heels, and the three enemies pursued him; while running, they got separated from each other because they were differently injured and they could run at different speeds. The first to reach the Orazio was the healthiest of the Curiazi, who was easily killed. Then, came the other two who were injured, and it was easy for the Orazio to kill them one by one.

This idea of “killing” one member at a time was applied to the three elements in the exact sequence of an ideal sheaf (together with the ideals of a residual scheme and a “trace”) by A. Hirschowitz in <cit.> (that is why now, we keep the french version “Horace” for Orazi) to compute the postulation of multiple points and count how many conditions they impose.


Even if the following definition extends to any scheme of fat points, since it is the case of our interest, we focus on the scheme of two-fat points.

We say that a scheme Z of r two-fat points, defined by the ideal I_Z, imposes independent conditions on the space of hypersurfaces of degree d in n+1 variable 𝒪_ℙ^n(d) if codim_(I_Z)_d) in S^dV is min{n+d  d , r(n+1)}.


This definition, together with the considerations of the previous section and (<ref>) allows us to reformulate the problem of finding the dimension of secant varieties to Veronese varieties in terms of independent conditions imposed by a zero-dimensional scheme of double fat points to forms of a certain degree.


 The s-th secant variety σ_s(X_n,d) of a Veronese variety has the expected dimension if and only if a scheme of s generic two-fat points in ℙ^n imposes independent conditions on 𝒪_ℙ^n(d).



The linear system ℒ := ℒ_n,2(-∑_i=1^s 2P_i) is special if s≤ n. Actually, quadrics in ℙ^n singular at s independent points P_1 , … , P_s are cones with the vertex ℙ^s-1 spanned by P_1 , … , P_s. Therefore, the system is empty as soon as s≥ n + 1, whereas, if s≤ n, one easily computes:

    (ℒ) = (ℒ) + s 2 .

Therefore, by (<ref>), this equality corresponds to the fact that σ_s(ν_2(ℙ^n)) are defective for all s≤ n; see Theorem <ref> (1).


We can now present how Alexander and Hirschowitz used the Horace method in <cit.> to compute the dimensions of the secant varieties of Veronese varieties.


Let Z ⊂ℙ^n be a scheme of two-fat points whose ideal sheaf is _Z. Let H⊂ℙ^n be a hyperplane. We define the following: 

	
  * the trace of Z with respect to H is the scheme-theoretic intersection: 
    _H(Z) := Z ∩ H;
 
	
  * the residue of Z with respect to H is the zero-dimensional scheme defined by the ideal sheaf _Z : 𝒪_ℙ^n(-H) and denoted _H(Z).




Let Z = 2P_0 ⊂ℙ^n be the two-fat point defined by ℘^2 = (x_1,…,x_n)^2, and let H be the hyperplane {x_n = 0}. Then, the residue _H(Z)⊂ℙ^n is defined by: 

    I__H(Z) = ℘^2 : (x_n) = (x_1,…,x_n) = ℘,

hence, it is a simple point of ℙ^n; the trace _H(Z)⊂ H ≃ℙ^n-1 is defined by: 

    I__H(Z) = (x_1,…,x_n)^2 ⊗[x_0,…,x_n] / (x_n) = (x_1,…,x_n-1)^2,

where the x_i's are the coordinate of the ^n-1≃ H, i.e., _H(Z) is a two-fat point in ^n-1 with support at P_0 ∈ H. 

The idea now is that it is easier to compute the conditions imposed by the residue and by the trace rather than those imposed by the scheme Z; in particular, as we are going to explain in the following, this gives us an inductive argument to prove that a scheme Z imposes independent conditions on hypersurfaces of certain degree. In particular, for any d, taking the global sections of the restriction exact sequence:

    0→ℐ__H(Z)(d-1)→ℐ_Z(d) →ℐ__H(Z)(d)→ 0,

we obtain the so-called Castelnuovo exact sequence:

    0→ (I__H(Z))_d-1→ (I_Z)_d→ (I__H(Z))_d,

from which we get the inequality:


    _(I_Z)_d ≤_(I__H(Z))_d-1 + _(I__H(Z))_d.

Let us assume that the supports of Z are r points such that t of them lie on the hyperplane H, i.e., _H(Z) is the union of r-t many two-fat points and t simple points in ^n and _H(Z) is a scheme of t many two-fat points in ^n-1 i.e., with the notation of linear systems introduced above,

    _(I__H(Z))_d-1    = (_n,d-1(-2∑_i=1^r-tP_i - ∑_i=r-t+1^rP_i)) + 1; 
    _(I__H(Z))_d     = (_n-1,d(-2∑_i=1^tP_i)) + 1.
 Assuming that:


  *  _H(Z) imposes independent conditions on 𝒪_ℙ^n(d-1), i.e.,

    _(I__H(Z))_d-1 = max{d-1+n  n -(r-t)(n+1)-t,0},


  *  and _H(Z) imposes independent conditions on 𝒪_ℙ^n-1(d), i.e.,

    _ (I__H(Z))_d = max{d+n-1  n+1 -tn,0},


then, by (<ref>) and since the expected dimension (Definition <ref>) is always a lower bound for the actual dimension, we conclude the following.


Let Z be a union of r many two-fat points in ℙ^n, and let H⊂ℙ^n be a hyperplane such that t of the r points of Z have support on H.
Assume that _H(Z_r) imposes independent conditions on 𝒪_H(d) and that _HZ_r imposes independent conditions on 𝒪_ℙ^n(d-1).
If one of the pairs of the following inequalities occurs:


  *  tn≤d+n-1n-1 and r(n+1)-tn≤d+n-1n,

  *  tn≥d+n-1n-1 and r(n+1)-tn≥d+n-1n,

then Z imposes independent conditions on the system 𝒪_ℙ^n(d).















The technique was used by Alexander and Hirschowitz to compute the dimension of the linear system of hypersurfaces with double base points, and hence, the dimension of secant varieties of Veronese varieties is mainly the Horace method, via induction. 

	The regularity of secant varieties can be proven as described above by induction, but non-regularity cannot. Defective cases have to be treated case by case. We have already seen that the case of secant varieties of Veronese surfaces (Example <ref>) and of quadrics (Example <ref>) are defective, so we cannot take them as the first step of the induction.





Let us start with σ_s(X_3,3)⊂ℙ^19. The expected dimension is 4s-1. Therefore, we expect that σ_5(X_3,3) fills up the ambient space. Now, let Z be a scheme of five many two-fat points in general position in ^3 defined by the ideal I_Z = ℘_1^2∩…∩℘_5^2. Since the points are in general position, we may assume that they are the five fundamental points of ^3 and perform our computations for this explicit set of points. Then, it is easy to check that:

    (S/℘_1^2∩⋯∩℘_5^2,3) = 19-_(I_Z)_3=19-0.

 Hence, σ_5(X_3,3) = ℙ^19, as expected. This implies that: 

    (σ_s(X_3,3))  is the expected one for all  s≤ 5.
 
Indeed, as a consequence of the following proposition, if the s-th secant variety is regular, so it is the (s-1)-th secant variety.

 Assume that X is s-defective and that σ_s+1(X)≠ℙ^N. Then, X is also (s+1)-defective.


 
Let δ_s be the s-defect of X. By assumptions and by Terracini's lemma, if P_1,…,P_s∈ X are general points, then the span T_P_1,…,P_s := ⟨ T_P_1X,…,T_P_sX⟩, which is the tangent space at a general point of σ_s(X), has projective dimension min(N, sn+s-1)-δ_s. Hence, adding one general point P_s+1, the space T_P_1,…,P_s, P_s+1, which is the span of
T_P_1,…,P_s and T_P_s+1X, has dimension at most min{N,sn+s-1}-δ_s+n+1.
This last number is smaller than N, while it is clearly smaller than (s+1)n+s. Therefore, X is (s+1)-defective.






In order to perform the induction on the dimension, we would need to study the case of d=4, s=8 in ℙ^3, i.e., σ_8(X_3,4). We need to compute _Z(4) = ([x_0, … , x_3]/(℘_1^2 ∩⋯∩℘_8^2), 4). In order to use the Horace lemma, we need to know how many points in the support of scheme Z lie on a given hyperplane. The good news is upper semicontinuity, which allows us to specialize points on a hyperplane. In fact, if the specialized scheme has the expected Hilbert function, then also the general scheme has the expected Hilbert function (as before, this argument cannot be used if the specialized scheme does not have the expected Hilbert function: this is the main reason why induction can be used to prove the regularity of secant varieties, but not the defectiveness). In this case, we choose to specialize four points on H, i.e., Z = 2P_1+…+2P_8 with P_1,…,P_4 ∈ H. Therefore,

	
  * _H(Z)=P_1+⋯ + P_4+ 2P_5+ ⋯ + 2P_8 ⊂^3;
	
  * _H(Z_8)=2 P_1 + ⋯ +2P_4⊂ H, where 2P_i's are two-fat points in ^2


Consider Castelunovo Inequality (<ref>). Four two-fat points in ℙ^3 impose independent conditions to 𝒪_ℙ^3(3) by (<ref>), then adding four simple general points imposes independent conditions; therefore, _HZ imposes the independent condition on 𝒪_ℙ^3(3). Again, assuming that the supports of _H(Z) are the fundamental points of ^2, we can check that it imposes the independent condition on 𝒪_ℙ^2(4). Therefore, 

    max{4+3  3 - 8 · 4,0}    = 3 = _ (I_Z)_4 ≤_ (I_Z)_4 
       ≤_ (I__H(Z))_3 + _ (I__H(Z))_4 
        = max{3+3  3 - 4· 4 - 4,0} + max{2+4  2 - 4 · 3,0}
        = max{20 - 16 - 4,0} + max{15-12,0} = 3.

In conclusion, we have proven that

    σ_s(X_3,4)  has the expected dimension for any  s≤ 8.


Now, this argument cannot be used to study σ_9(X_3,4) because it is one of the defective cases, but we can still use induction on d.


In order to use induction on the degree d, we need a starting case, that is the case of cubics. We have done ℙ^3 already; see (<ref>). Now, d=3, n=4, s=7 corresponds to a defective case. Therefore, we need to start with d=3 and n=5. We expect that σ_10(X_5,3) fills up the ambient space. Let us try to apply the Horace method as above. The hyperplane H is a ℙ^4; one two-fat point in ℙ^4 has degree five, so we can specialize up to seven points on H (in ℙ^4, there are exactly 35=7× 5 cubics), but seven two-fat points in ℙ^4 are defective in degree three; in fact, if Z = 2P_1+…+2P_7⊂^4, then _(I_Z)_3=1. Therefore, if we specialize seven two-fat points on a generic hyperplane H, we are “not using all the room that we have at our disposal”, and (<ref>) does not give the correct upper bound. In other words, if we want to get a zero in the trace term of the Castelunovo exact sequence, we have to “add one more condition on H”; but, to do that, we need a more refined version of the Horace method.









  §.§.§ La méthode d'Horace Differentielle


The description we are going to give follows the lines of <cit.>. 


An ideal I in the algebra of formal functions [[x,y]], where x = (x_1,… ,x_n-1), is called a 
vertically-graded (with respect to y) ideal if: 

    I = I_0 ⊕ I_1y ⊕⋯⊕ I_m-1y^m-1⊕ (y^m)

where, for i = 0,… ,m-1, I_i⊂ [[x]] is an
ideal.


Let Q be a smooth n-dimensional integral scheme, and let D be a
smooth irreducible divisor on Q. We say that Z ⊂ Q is a
vertically-graded subscheme of Q with base D and support
z∈ D, if Z is a zero-dimensional scheme with support at the point
z such that there is a regular system of parameters (x,y)
at z such that y=0 is a local equation for D and the ideal of
Z in 𝒪_Q,z≅ [[x,y]] is
vertically graded.


Let Z∈ Q be a vertically-graded subscheme with base D, and let p≥ 0
be a fixed integer. We denote by ^p_D(Z)∈ Q and
^p_D(Z)∈ D the closed subschemes defined, respectively, by
the ideals sheaves:

    ℐ_^p_D(Z) := ℐ_Z+
    (ℐ_Z:ℐ^p+1_D)ℐ^p_D,      and     ℐ_^p_D(Z),D := (ℐ_Z:ℐ^p_D)⊗𝒪_D.



In ^p_D(Z), we remove from Z the (p+1)-th “slice” of Z, while in
^p_D(Z), we consider only the (p+1)-th “slice”. Notice
that for p=0, this recovers the usual trace _D(Z) and residual schemes _D(Z).


Let Z⊂ℙ^2 be a three-fat point defined by ℘^3, with support at a point P ∈ H lying on a plane H ⊂ℙ^3. We may assume ℘ = (x_1,x_2) and H = {x_2 = 0}. Then, 3P is vertically graded with respect to H:


0.43

    ℘^3 = (x_1^3) ⊕ (x_1^2) x_2 ⊕ (x_1) x_2^2 ⊕ (x_2^3),


0.5


[line cap=round,line join=round,x=1.0cm,y=1.0cm,scale = 0.5]
(-1.5,-1.) rectangle (4.,3.5);
[line width=1.pt,dash pattern=on 1pt off 2pt] (0.,-1.) – (0.,3.);
[color = black] (-0.5,3.0) node x_2;
[line width=1.pt,dash pattern=on 1pt off 2pt,domain=-2.:3.] plot(,(-0.-0.*)/-1.);
[color = black] (3.5,-0.5) node x_1;

[fill=black] (1.,0.) circle (7.5pt);
[fill=black] (0.,1.) circle (7.5pt);
[fill=black] (1.,1.) circle (7.5pt);
[fill=black] (0.,2.) circle (7.5pt);
[fill=black] (0.,0.) circle (7.5pt);
[fill=black] (2.,0.) circle (7.5pt);




Visualization of a three-fat point in ^2: each dot corresponds to a generator of the local ring, which is Artinian.




Now, we compute all residues (white dots) and traces (black dots) as follows:



0.7
Case p=0.   ^0_H(P)⊂^2 is defined by: 

    ℘^3 : (x_2) = ℘^2 = (x_1^2) ⊕ (x_1) x_2 ⊕ (x_2^2),

i.e., it is a two-fat point in ^2, while ^0_H(P) is defined by:

    ℘^3 = ℘^3 ⊗[[x_0,x_1,x_2]]/(x_2) = (x_1^3),

where x_0,x_1 are the coordinates on H, i.e., it is a three-fat point in ^1.

0.3


[line cap=round,line join=round,x=1.0cm,y=1.0cm,scale = 0.5]
(-1.5,-1.) rectangle (4.,3.5);
[line width=1.pt,dash pattern=on 1pt off 2pt] (0.,-1.) – (0.,3.);
[color = black] (-0.5,3.0) node x_2;
[line width=1.pt,dash pattern=on 1pt off 2pt,domain=-2.:3.] plot(,(-0.-0.*)/-1.);
[color = black] (3.5,-0.5) node x_1;

(1.,0.) circle (7.5pt);
[fill=black] (0.,1.) circle (7.5pt);
[fill=black] (1.,1.) circle (7.5pt);
[fill=black] (0.,2.) circle (7.5pt);
(0.,0.) circle (7.5pt);
(2.,0.) circle (7.5pt);








0.7
Case p=1.   ^1_H(3P)⊂^2 is a zero-dimensional subscheme of ^2 of length four given by: 

    ℘^3 + (℘^3 : (x_2^2)) x_2     = (x_2^2,x_1x_2,x_1^3) = 
        = (x_1^3) ⊕ (x_1)x_2 ⊕ (x_2)^2;

roughly speaking, we “have removed the second slice” of 3P; while, ^1_H(3P) is given by:

    (℘^3 : (x_2)) ⊗[[x_0,x_1,x_2]] / (x_2) = (x_1^2).


0.3


[line cap=round,line join=round,x=1.0cm,y=1.0cm,scale = 0.5]
(-1.5,-1.) rectangle (4.,3.5);
[line width=1.pt,dash pattern=on 1pt off 2pt] (0.,-1.) – (0.,3.);
[color = black] (-0.5,3.0) node x_2;
[line width=1.pt,dash pattern=on 1pt off 2pt,domain=-2.:3.] plot(,(-0.-0.*)/-1.);
[color = black] (3.5,-0.5) node x_1;

(1.,0.) circle (7.5pt);
[fill=black] (0.,1.) circle (7.5pt);
[fill=black] (1.,1.) circle (7.5pt);
[fill=black] (0.,2.) circle (7.5pt);
(0.,0.) circle (7.5pt);
[fill = black] (2.,1.) circle (7.5pt);








0.7
Case p=2.   ^2_H(P)⊂^2 is a zero-dimensional scheme of length five given by: 

    ℘^3 + (℘^3 : (x_2^3)) x_2^2     = (x_2^2,x_1^2x_2,x_1^3) = 
        = (x_1^3) ⊕ (x_1^2)x_2 ⊕ (x_2)^2;

roughly speaking, we “have removed the third slice” of 3P; while ^2_H(P) is given by:

    (℘^3 : (x_2^2)) ⊗[[x_0,x_1,x_2]] / (x_2) = (x_1).


0.3


[line cap=round,line join=round,x=1.0cm,y=1.0cm,scale = 0.5]
(-1.5,-1.) rectangle (4.,3.5);
[line width=1.pt,dash pattern=on 1pt off 2pt] (0.,-1.) – (0.,3.);
[color = black] (-0.5,3.0) node x_2;
[line width=1.pt,dash pattern=on 1pt off 2pt,domain=-2.:3.] plot(,(-0.-0.*)/-1.);
[color = black] (3.5,-0.5) node x_1;

[fill=black] (1.,2.) circle (7.5pt);
[fill=black] (0.,1.) circle (7.5pt);
[fill=black] (1.,1.) circle (7.5pt);
[fill=black] (0.,2.) circle (7.5pt);
(0.,0.) circle (7.5pt);
[fill=black] (2.,1.) circle (7.5pt);



























Finally, let Z_1,… ,Z_r∈ Q be vertically-graded subschemes with base D
and support z_i; let Z=Z_1∪⋯∪ Z_r, and set p=(p_1,… ,p_r)∈ N^r. We write:

    ^p_D(Z):= ^p_1_D(Z_1)∪⋯∪^p_r_D(Z_r),
      ^p_D(Z):= ^p_1_D(Z_1)∪⋯∪^p_r_D(Z_r).

We are now ready to formulate the Horace differential lemma.
 
Let H be a hyperplane in ℙ^n, and let
W⊂ℙ^ n be a zero-dimensional closed subscheme. Let Y_1,… ,Y_r,  Z_1,… ,Z_r be zero-dimensional irreducible
subschemes of ℙ^n such that Y_i≅ Z_i, i=1,… ,r, Z_i
has support on H and is vertically graded with base H, and the
supports of Y=Y_1∪⋯∪ Y_r and Z=Z_1∪⋯∪ Z_r are
generic in their respective Hilbert schemes. Let p=(p_1,… ,p_r)∈ N^r. Assume:


  * H^0(ℐ_Tr_HW∪ Tr^p_H(Z),H(d))=0 and

  * H^0(ℐ_Res_HW∪ Res^p_H(Z)(d-1))=0;
 
then,

    H^0(ℐ_W∪ Y (d))=0.



For two-fat points, the latter result can be rephrased as follows.


Let H⊂ℙ^n be a hyperplane, P_1 , … , P_r∈ℙ^n be generic points and W ⊂ℙ^n be a zero-dimensional scheme. Let Z = 2P_1+⋯ + 2P_r⊂ℙ^n, and let Z' = 2P_1'+…+2P_r' such that the P_i''s are generic points on H. Let D_2,H(P_i') = 2P_i'∩ H be zero-dimensional schemes in ^n. Hence, let: 

    Z   = _H(W) + D_2,H(P_1') + … + D_2,H(P_r') ⊂^n,    and 
    T   = _H(W) + P'_1+…+P'_r ⊂ H ≃^n-1.

Then, if the following two conditions are satisfied:

    degue:      _(I_Z)_d-1=0; 
    dime:      _(I_T)_d=0,

then, _ (I_W+Z)_d = 0. 

Now, with this proposition, we can conclude the computation of σ_10(X_5,3). Before Section <ref>, we were left with the problem of computing the Hilbert function in degree three of a scheme Z = 2P_1+ ⋯ + 2P_10 of ten two-fat points with generic support in ℙ^5: since a two-fat point in ^5 imposes six conditions, the expected dimension of (I_Z)_3 is zero. In this case, the “standard” Horace method fails, since if we specialize seven points on a generic hyperplane, we lose one condition that we miss at the end of the game. We apply the Horace differential method to this situation. Let P'_1,…,P'_8 be generic points on a hyperplane H ≃^4 ⊂^5. Consider:

    Z    = P'_1 + … + P'_7 + D_2,H(P_8) + 2P_9 + 2P_10⊂^5,    and 
    T    = 2P'_1 + … + 2P'_7 + P'_8 ⊂ H ≃^4.

Now, dime is satisfied because we have added on the trace exactly the one condition that we were missing. It is not difficult to prove that degue is also satisfied: quadrics through Z are cones with the vertex the line between P_9 and P_10; hence, the dimension of the corresponding linear system equals the dimension of a linear system of quadrics in ^4 passing through a scheme of seven simple points and two two-fat points with generic support. Again, such quadrics in ^4 are all cones with the vertex the line passing through the support of the two two-fat points: hence, the dimension of the latter linear system equals the dimension of a linear system of quadrics in ^3 passing through a set of eight simple points with general support. This has dimension zero, since the quadrics of ^3 are ten. In conclusion, we obtain that the Hilbert function in degree three of a scheme of ten two-fat points in ^5 with generic support is the expected one, i.e., by (<ref>), we conclude that σ_10(X_5,3) fills the ambient space.




  §.§.§ Summary of the Proof of the Alexander–Hirshowitz Theorem


We finally summarize the main steps of the proof of Alexander–Hirshowitz theorem (Theorem <ref>):

	
  * The dimension of σ_s(X_n,d) is equal to the dimension of its tangent space at a general point Q;
	
  * By Terracini's lemma (Lemma <ref>), if Q is general in ⟨ P_1, …, P_s ⟩, with P_1, … , P_s∈ X general points, then: 
    (T_Q σ_s(X))=(⟨ T_P_1X, …, T_P_sX⟩);

	
  * By using the apolarity action (see Definition <ref>), one can see that: 
	
    (⟨ T_P_1X_n,d, …, T_P_sX_n,d⟩)= (R/(℘_1^2 ∩⋯∩℘_s^2),d)-1,

	where ℘_1^2 ∩⋯∩℘_s^2 is the ideal defining the scheme of two-fat points supported by ^n corresponding to the P_i's via the d-th Veronese embedding;
	
  * Non-regular cases, i.e., where the Hilbert function of the scheme of two-fat points is not as expected, have to be analyzed case by case; regular cases can be proven by induction:
	
	
  *  The list of non-regular cases corresponds to defective Veronese varieties and is very classical; see Section <ref>, page history and <cit.> for the list of all papers where all these cases were investigated. We explained a few of them in Examples <ref>, <ref> and <ref>;
	
  *  The proof of the list of non-regular cases classically known is complete and can be proven by a double induction procedure on the degree d and on the dimension n (see Theorem <ref> and Proposition <ref>):
	
		
  * The starting step of the induction for the degree is d=3 since quadrics are defective (Example <ref>):
		
			
  * The first case to study is therefore X_3,3: in order to prove that σ_5(X_3,3)= ℙ^19 as expected (see page first:step:induction), the Horace method (Section <ref>) is introduced. 
		
		
  * The starting step of the induction for the dimension is n=5:
		
			
  * σ_8(X_3,4) has the expected dimension thanks to upper semicontinuity (see page upper:semicontinuity), so also the smallest secant varieties are regular (page sigma834);
			
  * σ_9(X_3,4) is defective (<cit.>);
		 	
  * Therefore, one has to start with σ_10(X_3,5), which cannot be done with the standard Horace method (see page fail:Horace), while it can be solved (see page win:diff:horace) by using the Horace differential method (Section <ref>).









 §.§ Algorithms for the Symmetric-Rank of a Given Polynomial


The goal of the second part of this section is to compute the symmetric-rank of a given symmetric tensor. Here, we have decided to focus on algorithms rather than entering into the details of their proofs, since most of them, especially the more advanced ones, are very technical and even an idea of the proofs would be too dispersive. We believe that a descriptive presentation is more enlightening on the difference among them, the punchline of each one and their weaknesses, rather than a precise proof.




  §.§.§ On Sylvester's Algorithm


In this section, we present the so-called Sylvester's algorithm (Algorithm <ref>). It is classically attributed to Sylvester, since he studied the problem of decomposing a homogeneous polynomial of degree d into two variables as a sum of d-th powers of linear forms and solved it completely, obtaining that the decomposition is unique for general polynomials of odd degree. 
The first modern and available formulation of this algorithm is due to Comas and Seiguer; see <cit.>.

Despite the “age” of this algorithm, there are modern scientific areas where it is used to describe very advanced tools; see <cit.> for the measurements of entanglement in quantum physics. The following description follows <cit.>.


If V is a two-dimensional vector space, there is a well-known
isomorphism between ⋀^d-r+1(S^dV) and S^d-r+1(S^rV); see <cit.>. In terms of projective algebraic varieties, this isomorphism allows us to view the (d-r+1)-th Veronese embedding of ℙ^r ≃ S^rV as the set
of (r-1)-dimensional linear subspaces of ℙ^d that are
r-secant to the rational normal curve. The description of this
result, via coordinates, was originally given by Iarrobino and Kanev; see <cit.>. Here, we follow the description appearing in <cit.> (Lemma 2.1). We use the notation G(k,W) for the Grassmannian of k-dimensional linear spaces in a vector space W and the notation (k,n) for the Grassmannian of k-dimensional linear spaces in ^n.

  
Consider the map
ϕ_r,d-r+1:ℙ^(S^rV)→ G(d-r+1,S^dV) that sends the projective class of F ∈ S^rV to the
(d-r+1)-dimensional subspace of S^dV made by the multiples of F, i.e.,

    ϕ_r,d-r+1([F]) = F· S^d-rV ⊂ S^dV.
 
Then, the following hold:

	
  * the image of ϕ_r,d-r+1, after the Pl'́ ucker embedding of G(d-r+1,S^dV) inside (⋀^d-r+1S^dV), is the (d-r+1)-th Veronese embedding of S^rV;
	
  * identifying G(d-r+1,S^dV) with (r-1, S^dV^*), the above Veronese variety is the set of linear spaces r-secant to a rational normal curve _d ⊂ S^dV^*.



For the proof, we follow the constructive lines of <cit.>, which we keep here, even though we take the proof as it is, since it is short and we believe it is constructive and useful.

 
Let {x_0,x_1} be the variables on V. Then, write
F=u_0x_0^r+u_1x_0^r-1x_1+…+u_rx_1^r ∈ S^rV. A basis of
the subspace of S^dV of forms of the type FH is given by:

    x_0^d-r F     = u_0x_0^d+ ⋯ +u_rx_0^d-rx_1^r, 
    
    	x_0^d-rx_1 F     =      u_0x_0^d-1x_1+⋯ +u_rx_0^d-r-1x_1^r+1, 
      ⋮           ⋱
    
    	x_1^d-r F     =            u_0x_0^rx_1^d-r+⋯ +u_rx_1^d.

The coordinates of these elements with respect to the standard monomial basis
{x_0^d,x_0^d-1x_1,…,x_1^d} of
S^dV are thus given by the rows of the following (r+1) × (d+1) matrix:

    ([   u_0   u_1     …   u_r     0     …     0     0;     0   u_0   u_1     …   u_r     0     …     0;     ⋮     ⋱     ⋱     ⋱     ⋱     ⋱     ⋮;     0     …     0   u_0   u_1     …   u_r     0;     0     …     0     0   u_0     … u_r-1   u_r ]).

The standard Plücker coordinates of the subspace
ϕ_r,d-r+1([F]) are the maximal minors of this matrix. It is
known (see for example <cit.>) that these minors form a basis of
[u_0,…,u_r]_d-r+1, so that the image of ϕ_r,d-r+1([F]) is indeed a
Veronese variety, which proves (1).

To prove (2), we recall some standard facts from <cit.>. Consider 
homogeneous coordinates z_0,…,z_d in
ℙ^(S^dV^*), corresponding to the dual basis of the basis 
{x_0^d,x_0^d-1x_1,…,x_1^d}. Consider
_d⊂ℙ^(S^dV^*), the standard rational normal
curve with respect to these coordinates. Then, the image of [F]
by ϕ_r,d-r+1 is precisely the r-secant space to _d
spanned by the divisor on _d induced by the zeros of F.
This completes the proof of (2).


The rational normal curve _d⊂ℙ^d is the d-th Veronese embedding of V ≃^1 inside ℙS^dV≃^d. Hence, a symmetric tensor F∈ S^dV has symmetric-rank r if and only if r is the minimum integer for which there exists a ℙ^r-1≃ℙ W⊂ℙ^ S^dV
 such that F∈ℙ W and ℙ W is r-secant to the rational normal curve _d⊂ℙ^ (S^dV) in r distinct points. Consider the maps:

    ℙ^ (S^rV) ϕ_r, d-r+1⟶ (d-r, ℙS^dV)
    α_r, d-r+1≃ (r-1, ℙ^  S^dV^*).

Clearly, we can identify ℙS^dV^* with ℙS^dV; hence, the
Grassmannian 𝔾(r-1,ℙS^dV^*) can be identified with 𝔾 (r-1, ℙS^dV). Now, by Lemma <ref>, a projective subspace ℙW of
ℙS^dV^* ≃ℙS^dV ≃ℙ^d is r-secant to _d⊂ℙS^dV in r distinct points
if and only if it belongs to Im(α_r, d-r+1∘ϕ_r,d-r+1) and the preimage of ℙ^ W via α_r,d-r+1∘ϕ_r,d-r+1 is a polynomial with r distinct roots. Therefore, a symmetric tensor F ∈ S^dV has symmetric-rank r if and only if r is the minimum integer for which the following two conditions hold:


  * F belongs to some ℙ^ W ∈Im(α_r, d-r+1∘ϕ_r,d-r+1 )⊂𝔾 (r-1, ℙS^dV),

  * there exists a polynomial F ∈ S^rV that has r distinct roots and such that
α_r, d-r+1 ( ϕ_r,d-r+1([F]) )=ℙ^ (W).

Now, let U be a (d-r)-dimensional linear subspace of S^dV. 




The proof of Lemma <ref> shows that ℙU belongs to the image of ϕ_r,d-r+1 if and only if there exist u_0, … , u_r∈ such that U=⟨ F_1, … ,
F_d-r+1⟩, where, with respect to the standard monomial basis ℬ = {x_0^d,x_0^d-1x_1,…,x_1^d} of S^dV, we have:

    F_1    =(u_0, u_1,… , u_r, 0, … ,
    0), 
    
    F_2    = (0, u_0, u_1,… , u_r, 0, … ,
    0), 
      ⋮         ⋮
    
    F_d-r+1    =(0, … , 0, u_0, u_1,… , u_r).

Let ℬ^* = {z_0, … , z_d} be the dual basis of ℬ with respect to the apolar pairing. Therefore, there exists a W⊂ S^dV such that ℙ W =α_r,d-r+1(ℙU) if and only if W=H_1∩⋯∩ H_d-r+1, and the H_i's are as follows:

    H_1:     u_0z_0+⋯ +u_rz_r =0; 
    
    	H_2 :           u_0z_1 +⋯ +u_rz_r+1 = 0; 
      ⋮               ⋱
    
    	H_d-r+1:                 u_0z_d-r+⋯ +u_rz_d = 0.

This is sufficient to conclude that F ∈ℙ^ S^dV belongs
to an (r-1)-dimensional projective subspace of ℙ^ S^dV
that is in the image of α_r, d-r+1∘ϕ_r,d-r+1
defined in (<ref>) if and only if there exist H_1, …
, H_d-r+1 hyperplanes in S^dV as above, such that F ∈
H_1∩…∩ H_d-r+1. Now, given F ∈ S^dV with coordinates (a_0, … ,a_d) with respect to the dual basis ℬ^*, we have that F ∈
H_1∩…∩ H_d-r+1 if and only if the following linear
system admits a non-trivial solution in the u_i's

    u_0a_0+⋯ +u_ra_r    =0 
       u_0a_1 +⋯ +u_ra_r+1    =0 
        ⋱
           u_0a_d-r+⋯ +u_ra_d    =0.

If d-r+1<r+1, this system admits an infinite number of solutions. If r≤ d/2, it admits a non-trivial solution if and only if all the maximal (r+1)-minors of the following catalecticant matrix (see Definition <ref>) vanish:

    [   a_0     ⋯   a_r;   a_1     ⋯ a_r+1;     ⋮     ⋮; a_d-r     ⋯   a_d ].

 
The dimension of σ_r(_d) is never defective, i.e., it is the minimum between 2r-1
and d. Actually, σ_r(_d)⊊ℙS^dV if and only if 1≤ r < ⌈d+1/2⌉. Moreover, an element [F] ∈ℙS^dV belongs to σ_r(_d) for 1≤ r < ⌈d+1/2⌉, i.e., _sym(F) = r, if and only if

Cat_r,d-r(F) 

does not have maximal rank. 
These facts are very classical; see, e.g., <cit.>.

Therefore, if we consider the monomial basis {d  i^-1 x_0^ix_1^d-i |  i = 0,…,d} of S^dV and write F = ∑_i=0^d d  i^-1 a_i x_0^ix_1^d-i, then we write the (i,d-i)-th catalecticant matrix of F as Cat_i,d-i(F) = (a_h+k)_h = 0,…,i 
 k = 0,…,d-i.













































































 
Compute the symmetric-rank and a minimal Waring decomposition of the polynomial 
    F = 2x_0^4-4x_0^3x_1+30x_0^2x_1^2-28x_0x_1^3+17x_1^4.
 
We follow Sylvester's algorithm. The first catalecticant matrix with rank smaller than the maximal is:

    Cat_2,2(F)=
    [  2 -1  5; -1  5 -7;  5 -7 17 ],

in fact, rk Cat_2,2(F)=2. Now, let {y_0,y_1} the dual basis of V^*. We get that Cat_2,2(F)=⟨ 2y_0^2-y_0y_1-y_1^2 ⟩. We factorize: 

    2y_0^2-y_0y_1-y_1^2=(-y_0+y_1)(-2y_0-y_1).
 
Hence, we obtain the roots {(1,1),(1,-2)}. Then, it is direct to check that:

    [  1  1;  1 -2;  1  4;  1 -8;  1 16;    ][ 1; 1 ] =
    	[  2; -1;  5; -7; 17 ],

hence, a minimal Waring decomposition is given by: 

    F = (x_0+x_1)^4+(x_0-2x_1)^4.



The following result was proven by Comas and Seiguer in
 <cit.>; see also <cit.>. It describes the structure of the stratification by symmetric-rank of
symmetric tensors in S^dV with V=2. This result allows us to improve the classical Sylvester algorithm (see Algorithm <ref>).



Let _d = {[F] ∈ℙ S^dV  | _sym (F) =1} = {[L^d]  |  L ∈ S^1V}⊂ℙ^d be the rational normal curve of degree d parametrizing decomposable symmetric tensors. Then,

    ∀  r, 2≤ r≤⌈d+1 2⌉,       σ_r(_d) ∖σ_r-1(_d) = σ_r,r(_d)∪σ_r, d-r+2(_d),

where we write:

    σ_r,s(_d) := {[F] ∈σ_r(_d)  | _sym(F) = s}⊂σ_r(_d).











 
Let F=5x_0^5x_1, and let {y_0,y_1} be the dual basis to {x_0,x_1}. The smallest catalecticant without full rank is: 

    Cat_2,3(F)=
    [ 0 1 0 0; 1 0 0 0; 0 0 0 0 ],

which has rank two. Therefore [F]∈σ_2(C_6). Now, Cat_2,3(F)=⟨ y_1^2⟩, which has a double root. Hence, [F]∈σ_2,6(C_6).



When a form F ∈[x_0,… ,x_n] can be written using less
variables, i.e., F ∈[L_0,…,L_m], for L_j∈[x_0,…
,x_n]_1, with m<n, we say that F has m essential variables (in the literature, it is also said that F is m-concise). That is, F ∈ S^dW, where W = ⟨ L_0,…,L_m ⟩⊂ V. Then, the rank of [F] with respect to X_n,d is the same one as the one with respect to ν_d(ℙW) ⊂ X_n,d; e.g., see <cit.>. As recently clearly described in <cit.> and more classically in <cit.>, the number of essential variables of F coincides with the rank of the first catalecticant matrix Cat_1,d-1(F). In particular, when [F] ∈σ_r(X_n,d) ⊂ℙ (S^dV) with (V) = n+1, then, if
r < n+1, there is a subspace W⊂ V with (W) = r such
that [F] ∈ℙ S^dW, i.e., F can be written with respect to r variables.


Let now V be (n+1)-dimensional, and consider the following construction:

    [         Hilb_r(ℙ^ n)                    ϕ     𝔾(d+n  n-r,S^dV);                                                 [origin=c]-90≅;       𝔾(r-1,ℙS^dV^*)                    ⟵ 𝔾(d+n  n-r-1,ℙ S^dV) ]

where the map ϕ in (<ref>) sends a zero-dimensional scheme Z 
with (Z)=r to the vector space (I_Z)_d (it is defined in
the open set formed by the schemes Z, which impose independent
conditions to forms of degree d) and where the last arrow is the identification, which sends a linear space to its perpendicular. 

As in the case n=1, the final image from the latter construction
gives the (r-1)-spaces, which are r-secant to the Veronese
variety in ℙ^ N ≅ℙ ([x_0,…,x_n]_d)^*. Moreover, each
such space cuts the image of Z via the Veronese embedding.

 From now on, we will always use the notation Π_Z to indicate
 the projective linear subspace of dimension r-1 in ℙ S^dV, with (V)=n+1,
 generated by the image of a zero-dimensional scheme Z⊂ℙ^ n of degree r via the Veronese embedding, i.e., Π_Z = ⟨ν_d(Z) ⟩⊂ S^dV.


 Any [F] ∈σ_2(X_n,d)⊂ℙ S^dV,
 with (V)=n+1 can only have symmetric-rank equal to 1, 2 or d. More precisely:

    σ_2(X_n,d) ∖ X_n,d = σ_2,2(X_n,d)∪σ_2,d(X_n,d);

more precisely, σ_2,d(X_n,d) = τ (X_n,d)∖ X_n,d, where τ(X_n,d) denotes the tangential variety of X_n,d, i.e., the Zariski closure of the union of the tangent spaces to X_n,d.




This is actually a quite direct consequence of Remark <ref> and of Theorem <ref>, but let us describe the geometry in some detail, following the proof of <cit.>.
 Since r=2, every Z∈ Hilb_2(ℙ^ n) is the complete intersection of a line and a quadric, so the structure of I_Z is well known, i.e., I_Z =
(L_0,…,L_n-2,Q), where L_i's are linearly independent linear forms and Q is a quadric in S^2V ∖ (L_0,…,L_n-2)_2.

If F ∈σ_2(X_n,d), then we have two possibilities: either
_sym (F) =2

or _sym (T)>2, i.e., F lies on a tangent line Π_Z to the Veronese, which is
given by the image of a scheme Z ⊂ V of degree 2, via the maps
(<ref>). We can view F in the projective linear space H
≅ℙ^ d in ℙ (S^dV) generated by the rational normal curve
_d ⊂ X_n,d, which is the image of the line ℓ defined by
the ideal (L_0,…,L_n-2) in ℙV, i.e., ℓ⊂ℙ^n is the unique line containing Z. Hence, we can apply Theorem <ref> in order to get that _sym (F)≤ d. Moreover, by Remark <ref>, we have _sym(F)=d.


 
 Let us check that σ_2(X_n,d) is given by the annihilation of the (3× 3)-minors of the
first two catalecticant matrices, Cat_1,d-1(V) and Cat_2,d-2(V) (see Definition <ref>); actually, such minors are the
generators of I_σ_2(ν_d(ℙ^ n)); see <cit.>.

Following the construction above (<ref>), we can
notice that the coefficients of the linear spaces defined by the forms L_i∈ V^* in the
ideal I_Z are the solutions of a
linear system whose matrix is given by the catalecticant matrix Cat_1,d-1(V);
since the space of solutions has dimension n-1, we get
rk Cat_1,d-1(V) = 2. When we consider the quadric Q in I_Z, instead,
the analogous construction gives that its coefficients are the
solutions of a linear system defined by the catalecticant matrix Cat_2,d-2(V), and
the space of solutions give Q and all the quadrics in
(L_0,…,L_n-2)_2, which are n  2+2n-1, hence: 
    rk Cat_2,d-2(V) = n+2  2-(n  2+2n) = 2.



Therefore, we can write down an algorithm (Algorithm <ref>) to test if an element [F] ∈σ_2(X_n,d) has symmetric rank two or d.





 
Compute the symmetric-rank of 
    F = x_0^3x_2 + 3x_0^2x_1x_2 + 3x_0x_1^2x_2 + x_1^3x_2.
 
First of all, note that (y_0-y_1) ∘ F = 0; in particular, Cat_1,3(F) = ⟨ y_0-y_1 ⟩. Hence, F has two essential variables. This can also be seen by noticing that F = (x_0+x_1)^3x_2. Therefore, if we write z_0 = x_0 + x_1 and z_1 = x_2, then F = z_0^3z_1 ∈[z_0,z_1]. Hence, we can apply Algorithm <ref> and <ref> to compute the symmetric-rank, symmetric-border rank and a minimal decompositions of F. 











In particular, we write: 

    Cat_2,2(F)=
    [   0 1/4   0; 1/4   0   0;   0   0   0 ],

which has rank two, as expected. Again, as in Example <ref>, the kernel of Cat_2,2(F) defines a polynomial with a double root. Hence, _sym(F) = 2 and _sym(F) = 4. If we are interested in finding a minimal decomposition of F, we have to consider the first catalecticant whose kernel defines a polynomial with simple roots. In this case, we should get to:

    Cat_0,4(F) =
    	[   0; 1/4;   0;   0;   0 ],

whose kernel is ⟨ (1,0,0,0,0),(0,0,1,0,0), (0,0,0,1,0), (0,0,0,0,1) ⟩. If we let {w_0,w_1} be the variables on W^*, we take a polynomial in this kernel, as for example G = w_0^4+w_0^2w_1^2+w_0w_1^3+w_1^4. Now, if we compute the roots of G, we find four complex distinct roots, i.e.,

    [  (α_1,β_1) = -1/6A - 1/2√(B+C) (α_2,β_2) = -1/6A + 1/2√(B+C);;   (α_3,β_3) = 1/6A - 1/2√(B-C)  (α_4,β_4) = 1/6A + 1/2√(B-C); ]

where:

    A     =√(9(1/18 i√(257)√(3) - 43/54)^2/3 - 6 (1/18 i √(257)√(3) - 43/54)^1/3 + 13/1/18 i √(257)√(3) - 43/54^1/3); 
    
    
    B     = 
    √(-(1/18 i √(257)√(3) - 43/54)^1/3 -( 13/9/1/18 i √(257)√(3) - 43/54)^1/3); 
    
    
    C     = 6/√(9 (1/18 i √(257)√(3) - 43/54)^2/3 - 6 (1/18 i √(257)√(3) - 43/54)^1/3 + 13/1/18 i √(257)√(()3) - 43/54^1/3) - 4/3


Hence, if we write L_i = α_iz_0+β_iz_1, for i = 1,…,4, we can find suitable λ_i's to write a minimal decomposition F = ∑_i=1^4 λ_i L_i^4. 
Observe that any hyperplane through [F] that does not contain the tangent line to _4 at [z_0^4] intersects C_4 at four distinct points, so we could have chosen also another point in ⟨ (1,0,0,0,0),(0,0,1,0,0), (0,0,0,1,0), (0,0,0,0,1) ⟩, and we would have found another decomposition of F.


Everything that we have done in this section does not use anything more than Sylvester's algorithm for the two-variable case. In the next sections, we see what can be done if we have to deal with more variables and we cannot reduce to the binary case like in Example <ref>.


Sylvester's algorithm allows us to compute the symmetric-rank of any polynomial in two essential variables. It is mainly based on the fact that equations for secant varieties of rational normal curves are well known and that there are only two possibilities for the symmetric-rank of a given binary polynomial with fixed border rank (Theorem <ref>). Moreover, those two cases are easily recognizable by looking at the multiplicity of the roots of a generic polynomial in the kernel of the catalecticant.

The first ideas that were exploited to generalize Sylvester's result to homogeneous polynomials in more than two variables were:

	
  * a good understanding of the inverse system (and therefore, of the scheme defined by the kernels of catalecticant matrices and possible extension of catalecticant matrices, namely Hankel matrices); we will go into the details of this idea in Section <ref>;
	
  * a possible classification of the ranks of polynomials with fixed border rank; we will show the few results in this direction in Section <ref>.








  §.§.§ Beyond Sylvester's Algorithm Using Zero-Dimensional Schemes


We keep following <cit.>. Let us start by considering the case of a homogeneous polynomial with three essential variables.

If [F]∈σ_3(ν_d(ℙ^n))∖σ_2(ν_d(ℙ^n)), then we will need more than two variables, but actually, three are always sufficient. In fact, if [F] ∈σ_3(ν_d(ℙ^n)), then there always exists a zero-dimensional scheme ν_d(Z) of length three contained in ν_d(ℙ^n), whose span contains [F]; the scheme Z⊂ℙ^n itself spans a ℙ^2, which can be seen as ℙ((L_1,L_2,L_3)_1) with L_i's linear forms. Therefore, F can be written in three variables. The following theorem computes the symmetric-rank of any polynomial in [F] ∈σ_3(ν_d(ℙ^n))∖σ_2(ν_d(ℙ^n)), and the idea is to classify the symmetric-rank by looking at the structure of the zero-dimensional scheme of length three, whose linear span contains [F].

 Let d≥ 3, X_n,d⊂ℙ(^n+1). Then,

    σ_3(X_n,3)∖σ_2(X_n,3) =σ_3,3(X_n,3)∪σ_3,4(X_n,3)∪σ_3,5(X_n,3),
 
while, for d≥ 4,

    σ_3(X_n,d)∖σ_2(X_n,d)=σ_3,3(X_n,d)∪σ_3,d-1
    (X_n,d)∪σ_3,d+1(X_n,d)∪σ_3,2d-1(X_n,d).










































































































We do not give here all the details of the proof since they can be found in <cit.>; they are quite technical, but the main idea is the one described above. We like to stress that the relation between the zero-dimensional scheme of length three spanning F and the one computing the symmetric-rank is in many cases dependent on the following Lemma <ref>. Probably, it is classically known, but we were not able to find a precise reference. 


 Let Z ⊂ℙ^n, n≥ 2,
be a zero-dimensional scheme, with (Z) ≤ 2d+1. A
necessary and sufficient condition for Z to impose independent
conditions on hypersurfaces of degree d is that no line
ℓ⊂ℙ^ n is such that (Z∩ℓ) ≥ d+2.










































 Notice that if (ℓ∩ Z) is exactly d+1+k, then
the dimension of the space of curves of degree d through them
is increased exactly by k with respect to the generic case.


It is easy to see that Lemma <ref> can be improved as follows; see <cit.>.

 Let Z ⊂ℙ^ n, n≥ 2,
be a zero-dimensional scheme, with (Z) ≤ 2d+1. If h^1(ℙ^n, ℐ_Z(d))>0, there exists a unique line
ℓ⊂ℙ^ n such that (Z∩ℓ) = d+1+ h^1(ℙ^n, ℐ_Z(d))>0.


We can go back to our problem of finding the symmetric-rank of a given tensor.
The classification of symmetric-ranks of the elements in σ_4(X_n,d) can be treated in an analogous way as we did for σ_3(X_n,d), but unfortunately, it requires a very complicated analysis on the schemes of length four. This is done in <cit.>, but because of the long procedure, we prefer to not present it here.

It is remarkable that σ_4(X_n,d) is the last s-th secant variety of Veronesean, where we can use this technique for the classification of the symmetric-rank with respect to zero-dimensional schemes of length s, whose span contains the given polynomial we are dealing with; for s≥ 5, there is a more intrinsic problem. In fact, there is a famous counterexample due to Buczyńska and Buczyśki (see <cit.>) that shows that, in σ_5(X_4,3), there is at least a polynomial for which there does not exist any zero-dimensional scheme of length five on X_4,3, whose span contains it. The example is the following.
[Buczyńska, Buczyński <cit.>]

One can easily check that the following polynomial: 

    F= x_0^2 x_2 + 6 x_1^2 x_3 -3  (x_0+x_1)^2
    x_4.

can be obtained as lim_ϵ→ 01 3ϵ F_ϵ =F where: 

    F_ϵ= (x_0+ϵ x_2)^3 + 6(x_1+ϵ
    x_3)^3 -3(x_0+x_1+ ϵ x_4)^3 + 3(x_0 + 2 
    x_1)^3 - (x_0 + 3 x_1)^3
 
has symmetric-rank five for ϵ >0. Therefore, [F]∈σ_5(ν_3(ℙ^4)).

An explicit computation of F^⊥ yields the 
Hilbert series for _R/F^(z) = 1+5z+5z^2+z^3. 
Let us prove, by contradiction, that there is no saturated ideal I⊂ F^⊥ defining a zero-dimensional scheme of length ≤ 5.
Suppose on the contrary that I is such an ideal. Then, _R/ I(i) ≥_R/ F^⊥(i) for all i ∈ℕ. As _R/ I(i) is
an increasing function of i∈ℕ with _R/F^⊥ (i) ≤_R/ I (i) ≤ 5, we deduce that 
_R/ I(t) = 1 + 5∑_i=1^∞ z^i. 
This shows that I_1={0}
and that I_2 = (F^⊥)_2. As I is saturated, I_2:
(x_0, …, x_4)=I_1={0}, since _R/F^⊥ (1) =
5. However, an explicit computation of
(F^)_2: (x_0, …, x_4) gives
⟨ x_2,x_3,x_4⟩. In this way, we obtain a contradiction, so that
there is no saturated ideal of
degree ≤ 5 such that I⊂ F^⊥.
Consequently, the minimal zero-dimensional scheme contained in X_4,3 whose linear span contains [F] has degree six.


In the best of our knowledge, the two main results that are nowadays available to treat these “wild” cases are the following.


 Let X⊂ℙ^ N be a non-degenerate smooth variety.
Let H_r be the irreducible component of the Hilbert scheme of
zero-dimensional schemes of degree r of X containing r distinct
points, and assume that for each y∈ H_r, the corresponding
subscheme Y of X imposes independent conditions on linear forms.
Then, for each P∈σ_r(X) ∖σ_r^0(X),
there exists a zero-dimensional scheme Z⊂ X of degree r such
that P∈⟨ Z⟩≅ℙ^ r-1. Conversely, if there exists Z∈ H_r such that P∈⟨ Z⟩, then P∈σ_r(X).













Obviously, five points on a line do not impose independent conditions on cubics in any ℙ^n for n≥ 5; therefore, this could be one reason why the counterexample given in Example <ref> is possible. Another reason is the following.

Suppose there exist points P_1, … ,P_r ∈ X that are
linearly degenerate, that is ⟨ P_1, …, P_r ⟩ < r -1. Then, the join of
the r tangent stars (see <cit.> (Section 1.4) for a definition) at these points is contained in σ_r(X). In the case that 
X is smooth at P_1, … P_r, then ⟨ T_P_1X,…,T_P_rX ⟩⊂σ_r(X).
 




  §.§.§ Beyond Sylvester's Algorithm via Apolarity


We have already defined in Section <ref> the apolarity action of S^∙ V^* ≃[y_0,…,y_n] on S^∙ V ≃[x_0,…,x_n] and inverse systems. Now, we introduce the main algebraic tool from apolarity theory to study ranks and minimal Waring decompositions: that is the apolarity lemma; see <cit.>. First, we introduce the apolar ideal of a polynomial.

	Let F ∈ S^dV be a homogeneous polynomial. Then, the apolar ideal of F is:
	
    F^⊥ = {G ∈ S^∙ V^*  |  G ∘ F = 0}.



The apolar ideal is a homogeneous ideal. Clearly, F^⊥_i = S^iV^*, for any i > d, namely A_F = S^∙ V^*/F^⊥ is an Artinian algebra with socle degree equal to d. Since _ (A_F)_d = 1, then it is also a Gorenstein algebra. Actually, Macaulay proved that there exists a one-to-one correspondence between graded Artinian Gorenstein algebras with socle degree d and homogeneous polynomials of degree d; for details, see <cit.>.


Note that, directly by the definitions, the non-zero homogeneous parts of the apolar ideal of a homogeneous polynomial F coincide with the kernel of its catalecticant matrices, i.e., for i = 0,…,d,

    F^⊥_i = (Cat_i,d-i(F)).


The apolarity lemma tells us that Waring decompositions of a given polynomial correspond to sets of reduced points whose defining ideal is contained in the apolar ideal of the polynomial.

 Let Z={[L_1], … , [L_r]}⊂ℙ(S^1V), then the following are equivalent:
 
 	
  * F=∑_i=1^r λ_i L_i^d, for some λ_1,…,λ_r ∈;
 	
  * I(Z)⊆ F^⊥.
 
 If these conditions hold, we say that Z is a set of points apolar to F.


The fact that (1) implies (2) follows from the easy fact that, for any G ∈ S^dV^*, we have that G ∘ L^d is equal to d times the evaluation of G at the point [L] ∈ V. Conversely, if I(Z) ⊂ F^⊥, then we have that F ∈ I(Z)^⊥_d = ⟨ L_1^d,…,L_r^d ⟩; see Remark <ref> and Proposition <ref>.

[Yet again: Sylvester's algorithm]
With this lemma, we can rephrase Sylvester's algorithm. Consider the binary form F =∑_i=0^d c_i d i x_0^d-ix_1^i. Such an F can be decomposed as the sum of r distinct powers of linear forms if and only if there exists Q =q_0y_0^r+q_1y_0^r-1y_1+ ⋯ + q_ry_1^r such that:

    ( [   c_0   c_1     ⋯   c_r;   c_1     ⋯ c_r+1;     ⋮     ⋮; c_d-r     ⋯   c_d ])( [ q_0; q_1;   ⋮; q_r ])=0,

and Q=μΠ_k=1^r (β_k y_0 - α_k y_1), for a suitable scalar μ∈, where [α_i:β_i]'s are different points in ^1. In this case, there exists a choice of λ_1,…,λ_r such that F =∑_k=1^r λ_k (α_k x_0 + β_k x_1)^d. This is possible because of the following remarks:

	
  * Gorenstein algebras of codimension two are always complete intersections, i.e., 
    I(Z)⊂ (F^⊥)=(G_1,G_2);

	
  * Artinian Gorenstein rings have a symmetric Hilbert function, hence:
	
    (G_1) + (G_2) = (F) + 2,   say (G_1) ≤(G_2);

	
  * If G_1 is square-free, i.e., has only distinct roots, we take Q = G_1 and _ sym(F) = (G_1); otherwise, the first degree where we get something square-free has to be the degree of G_2; in particular, we can take Q to be a generic element in F^⊥_(G_2) and _ sym(F) = (G_2).
 




By using Apolarity Theory, we can describe the following algorithm (Algorithm <ref>).



 Compute a Waring decomposition of: 

    F=3x^4 + 12x^2y^2 + 2y^4 - 12x^2yz + 12xy^2z - 4y^3z + 12x^2z^2 - 12xyz^2 + 6y^2z^2 - 4yz^3 + 2z^4.
 The most square catalecticant matrix is: 

    Cat_2,2(F)=([  3  0  0  2 -1  2;  0  2 -1  0  1 -1;  0 -1  2  1 -1  0;  2  0  1  2 -1  1; -1  1 -1 -1  1 -1;  2 -1  0  1 -1  2 ]).

Now, compute that the rank of Cat_2,2(F) is three, and its kernel is: 

    (Cat_2,2(F))     = ⟨(1,0,0,-1,-1,-1), (0,1,0,-1,-2,0), (0,0,1,0,2,1)⟩ =
       ⟨ y_0^2-y_1^2-y_1y_2-y_2^2, y_0y_1-y_1^2-2y_1y_2, y_0y_2+2y_1y_2+y_2^2 ⟩⊂ S^2V^*.
 
It is not difficult to see that these three quadrics define a set of reduced points {[1:1:0], [1:0:-1], [1:-1:1]}⊂ V. Hence, we take L_1=x_0+x_1, L_2=x_0-x_2 and L_3=x_0-x_1+x_2, and, by the apolarity lemma, the polynomial F is a linear combinations of those forms, in particular, 
    F=(x_0+x_1)^4+(x_0-x_2)^4+(x_0-x_1+x_2)^4.


Clearly, this method works only if _ sym(F) =  rank Cat_m,d-m(F), for m = ⌈d/2⌉. Unfortunately, in many cases, this condition is not always satisfied.



Algorithm <ref> has been for a long time the only available method to handle the computation of the decomposition of polynomials with more than two variables. In 2013, there was an interesting contribution due to Oeding and Ottaviani (see <cit.>), where the authors used vector bundle techniques introduced in <cit.> to find non-classical equations of certain secant varieties. In particular, the very interesting part of the paper <cit.> is the use of representation theory, which sheds light on the geometric aspects of this algorithm and relates these techniques to more classical results like the Sylvester pentahedral theorem (the decomposition of cubic polynomial in three variables as the sum of five cubes). For the heaviness of the representation theory background needed to understand that algorithm, we have chosen to not present it here. Moreover, we have to point out that <cit.> (Algorithm 4) fails whenever the symmetric-rank of the polynomial is too large compared to the rank of a certain matrix constructed with the techniques introduced in <cit.>, similarly as happens for the catalecticant method.



Nowadays, one of the best ideas to generalize the method of catalecticant matrices is due to Brachat, Comon, Mourrain and Tsidgaridas, who in <cit.> developed an algorithm (Algorithm <ref>) that gets rid of the restrictions imposed by the usage of catalecticant matrices. The idea developed in <cit.> is to use the so-called Hankel matrix that in a way encodes all the information of all the catalecticant matrices. The algorithm presented in <cit.> to compute a Waring decomposition of a form F ∈ S^dV passes through the computation of an affine Waring decomposition of the dehomogenization f of the given form with respect to a suitable variable. Let S = [x_1,…,x_n] be the polynomial ring in n variables over the field  corresponding to such dehomogenization. 


We first need to introduce the definition of Hankel operator associated with any Λ∈ S^*. To do so, we need to use the structure of S^* as the S-module, given by: 

    a * Λ : S →,    b ↦Λ(ab),    for  a ∈ S, Λ∈ S^*.

Then, the Hankel operator associated with Λ∈ S^* is the matrix associated with the linear map:

    H_Λ: S → S^*,  such that  a ↦ a * Λ.

Here are some useful facts about Hankel operators.
 
	(H_Λ) is an ideal.








Let I_Λ = (H_Λ) and A_Γ = S / I_Λ.
 

If rank(H_Λ)=r < ∞, then the algebra A_Λ is a -vector space of dimension r, and there exist polynomials l_1, … l_k of degree one and g_1, …, g_k of degree d_1,…,d_k, respectively, in [∂_1,…,∂_n] such that: 
    Λ = ∑_i=1^k l_i^d-d_ig_i.
 Moreover, I_Λ defines the union of affine schemes Z_1,…,Z_k with support on the points l^*_1,…,l^*_k ∈^n, respectively, and with multiplicity equal to the dimension of the vector space spanned by the inverse system generated by l_i^d-d_ig_i.


The original proof of this proposition can be found in <cit.>; for a more detailed and expanded presentation, see <cit.>.

 
An element Λ∈ S^* can be decomposed as Λ = ∑_i=1^r λ_il_i^d if and only if rank H_Λ = r, and I_Λ is a radical ideal.


Now, we consider the multiplication operators in A_Λ. Given a ∈ A_Λ:

    M_a: A_Λ→ A_Λ,


    b↦ a· b,

and, 

    M_a^t: A_Λ^* → A^*_Λ,


    γ↦ a * γ.

Now, 

    H_a*Λ := M_a^t · H_Λ.


 If A_Λ < ∞, then, Λ = ∑_i=1^k l_i^d-d_ig_i and: 


  * the eigenvalues of the operators M_a and M_a^t are given by {a(l^*_1), … , a(l^*_r)};

  * the common eigenvectors of the operators (M^t_x_i)_1≤ i ≤ n are, up to scalar, the l_i's.


Therefore, one can recover the l_i's, i.e., the points l^*_i's, by eigenvector computations: take B as a basis of A_f, i.e., say B = {b_1,…,b_r} with r= rank H_Λ, and let H^B_a * Λ=M_a^tH_Λ^B=H_Λ^BM_a (M_a is the matrix of the multiplication by a in the basis B). The common solutions of the generalized eigenvalue problem: 

    (H_a*Λ-λ H_Λ)v=0,

for all a∈ S yield the common eigenvectors H^B_Λ v of M_a^t, that is the evaluations at the points l_i^*'s. Therefore, these common eigenvectors H^B_Λ v are up to scalar the vectors [b_i(l^*_i), … , b_r(l^*_i)], for i = 1,…,r.

If f=∑_i=1^r λ_il_i^d, then the Z_i's in Proposition <ref> are simple, and one eigenvector computation is enough: in particular, for any a∈ S, M_a is diagonalizable, and the generalized eigenvectors H^B_Λ v are, up to scalar, the evaluations at the points l_i^*'s.


Now, in order to apply this algebraic tool to our problem of finding a Waring decomposition of a homogeneous polynomial F ∈[x_0,…,x_n], we need to consider its dehomogenization f = F(1,x_1,…,x_n) with respect to the variable x_0 (with no loss of generality, we may assume that the coefficients with respect to x_0 are all non-zero). Then, we associate a truncated Hankel matrices as follows.

Let B be a subset of monomials in S. We say that B is
connected to one if ∀  m∈ B either m=1 or there exists i∈{1,…,n} and m'∈ B such that m=x_i m'.

Let B,B' ⊂ S_≤ d be sets of monomials of degree ≤ d, connected to one.
For any f = ∑_α∈^n
 |α| ≤ d c_αd  d-|α|,α_1,…,α_n^α∈ S_d, we consider the Hankel matrix:

    H^B,B'_f = (h_α+β)_α∈ B, β∈ B',

where h_α = c_α if
|α|≤ d, and otherwise, h_α is an unknown.
The set of all these new variables is denoted h. Note that, by this definition, the known parts correspond to the catalecticant matrices of F. For simplicity, we write H^B_f = H^B,B_f. This matrix is also called quasi-Hankel <cit.>.


	Consider F = -4x_0x_1 + 2x_0x_2 +2x_1x_2 + x_2^2 ∈[x_0,x_1,x_2]. Then, we look at the dehomogenization with respect to x_0 given by f = -4x_1+2x_2+2x_1x_2 + x_2^2 ∈[x_1,x_2]. Then, if we consider the standard monomial basis of S_≤ 2 given by B = {1,x_1,x_2,x_1^2,x_1x_2,x_2^2}, then we get:
	
    H_f^B = [       0      -2       1       0       1       1;      -2       0       1 h_(3,0) h_(2,1) h_(1,2);       1       1       1 h_(2,1) h_(1,2) h_(0,3);       0 h_(3,0) h_(2,1) h_(4,0) h_(3,1) h_(2,2);       1 h_(2,1) h_(1,2) h_(3,1) h_(2,2) h_(1,3);       1 h_(1,2) h_(0,3) h_(2,2) h_(1,3) h_(0,4);         ],

	where the h's are unknowns.

Now, the idea of the algorithm is to find a suitable polynomial f whose Hankel matrix extends the one of f, has rank equal to the Waring rank of f and the kernel gives a radical ideal. This is done by finding suitable values for the unknown part of the Hankel matrix of f. Those f are elements whose homogenization is in the following set: 

    ℰ^d,0_r:=
    {[F]∈ℙ(S^dV)  | 1∃ L ∈ S^1V∖{ 0}, ∃F'∈ Y^m,m'_r s.t. L^m+m'-d F' = F
    1with m=max{ r, ⌈ d/2⌉}, m'=max{ r-1, ⌊ d/2 ⌋}}

where Y^i,d-i_r={[F]∈ℙ(S^dV)   |   rank Cat_i,d-i(F) ≤ r}.
If [F] ∈ℰ^d,0_r, we say that f is the generalized affine decomposition of size r.

Suppose that H^B,B'_f is invertible in (h), then we
define the formal multiplication operators: 

    M_i^B,B'(h) := (H^B,B'_f)^-1H^B,B'_x_if.


	If B is a subset of monomials, then we write B^+ = B ∪ x_1B ∪…∪ x_nB. Note that, if B is connected to on,e then also B^+ is connected to one.

The key result for the algorithm is the following.
 
If B and B' are sets of monomials connected to one, the coefficients of f are known on B^+× B'^+, and if H^B,B'_f̃ is invertible, then f extends uniquely to S if and only if: 

    M_i^B,B'· M_j^B,B'=M_j^B,B'· M_i^B,B',    for any 1 ≤ i < j ≤ n.







For simplicity, we give the example chosen by the authors of <cit.>.

 

We look for a decomposition of:

    F =     -1549440
     x_0x_1x_2^3 +2417040 x_0x_1^2x_2^2
      +166320 x_0^2x_1x_2^2-829440 x_0x_1^3x_2 
        -5760 x_0^3x_1x_2-222480 x_0^2x_1^2x_2
      +38 x_0^5-497664 x_1^5-1107804 x_2^5
        -120 x_0^4x_1+180 x_0^4x_2+12720 x_0^3x_1^2
      +8220 x_0^3x_2^2-34560 x_0^2x_1^3
        -59160 x_0^2x_2^3+831840 x_0x_1^4+442590 x_0x_2^4-5591520 x_1^4x_2 
        +7983360 x_1^3x_2^2-9653040 x_1^2x_2^3+5116680 x_1x_2^4.



  * We form a n+d-1  d×n+d-1  d matrix,
 the rows and the columns of which correspond to the coefficients
 of the polynomial with respect to the expression f = F(1,x_1,…,x_n) = ∑_α∈^n 
 |α| ≤ d c_αd  d-|α|,α_1,…,α_n^α.
 
The whole 21 × 21 matrix is the following.
  


  (
   [                   1       x_1       x_2     x_1^2   x_1 x_2     x_2^2     x_1^3  x_1^2x_2 x_1 x_2^2     x_2^3;         1        38       -24        36      1272      -288       822     -3456     -7416      5544     -5916;       x_1       -24      1272      -288     -3456     -7416      5544    166368    -41472     80568    -77472;       x_2        36      -288       822     -7416      5544     -5916    -41472     80568    -77472     88518;     x_1^2      1272     -3456     -7416    166368    -41472     80568   -497664  -1118304    798336   -965304;   x_1 x_2      -288     -7416      5544    -41472     80568    -77472  -1118304    798336   -965304   1023336;     x_2^2       822      5544     -5916     80568    -77472     88518    798336   -965304   1023336  -1107804;     x_1^3     -3456    166368    -41472   -497664  -1118304    798336   h_6,0,0   h_5,1,0   h_4,2,0   h_3,3,0; x_1^2 x_2     -7416    -41472     80568  -1118304    798336   -965304   h_5,1,0   h_4,2,0   h_3,3,0   h_2,4,0; x_1 x_2^2      5544     80568    -77472    798336   -965304   1023336   h_4,2,0   h_3,3,0   h_2,4,0   h_1,5,0;     x_2^3     -5916    -77472     88518   -965304   1023336  -1107804   h_3,3,0   h_2,4,0   h_1,5,0   h_0,6,0 ])

	

  Notice that we do not know the elements in some positions of the matrix. 
  In this case, we do not know the elements that correspond to monomials with (total) 
  degree higher than five.


  * We extract a principal minor of full rank.
 
 We should re-arrange the rows and the columns of the matrix so that there
 is a principal minor of full rank. We call this minor Δ_0. In
 order to do that, we try to put the matrix in row echelon form, using
 elementary row and column operations.
 
  In our example, the 4 × 4 principal minor is of full rank, 
  so there is no need for re-arranging the matrix.
  The matrix Δ_0 is: 
  
   Δ_0 = 
   (
   [     38    -24     36   1272;    -24   1272   -288  -3456;     36   -288    822  -7416;   1272  -3456  -7416 166368 ] 
   )
  
  Notice that the columns of the matrix correspond to the set of monomials
  { 1, x_1, x_2, x_1^2}.
 
 
  * We compute the “shifted” matrix Δ_1 = x_1 Δ_0.

 The columns of Δ_0 correspond to the set of some monomials, 
 say x^α, where α⊂ℕ^n.
 The columns of Δ_1 correspond to the set of monomials
 x_1  x^α.
 
  The shifted matrix Δ_1 is: 
  
   Δ_1 = 
   (
   [     -24    1272    -288   -3456;    1272   -3456   -7416  166368;    -288   -7416    5544  -41472;   -3456  166368  -41472 -497664 ] 
   ) .
  
  Notice that the columns correspond to the monomials
  { x_1, x_1^2, x_1 x_2, x_1^3}, 
  which are just the corresponding monomials of the columns of Δ_0,
  i.e., { 1, x_1, x_2, x_1^2}, multiplied by x_1.
 
 In this specific case, all the elements of the matrices Δ_0 and
 Δ_1 are known. If this is not the case, then we can compute the unknown entries
 of the matrix, using either necessary or sufficient conditions of the quotient algebra,
 e.g., it holds that the M_x_iM_x_j - M_x_jM_x_i = 0, for any i, j ∈{1, …, n}.
 
 

  * We solve the equation (Δ_1 - λΔ_0) X = 0.

 We solve the generalized eigenvalue/eigenvector problem <cit.>.
 We normalize the elements of the eigenvectors so that the first element is one, 
 and we read the solutions from the coordinates of the normalized eigenvectors.
 
  The normalized eigenvectors of the generalized eigenvalue problem are:
  
   ( [   1; -12;  -3; 144 ] ) ,
   ( [   1;  12; -13; 144 ] ) , 
   ( [  1; -2;  3;  4 ] ) , 
   ( [ 1; 2; 3; 4 ]
   ).
  


  The coordinates of the eigenvectors correspond to the elements of the monomial basis
  { 1, x_1, x_2, x_1^2}.
  Thus, we can recover the coefficients of x_1 and x_2 
  in the decomposition from the coordinates of the eigenvectors.

  Recall that the coefficients of x_0 are considered to be one because of the dehomogenization process. Thus, our polynomial admits a decomposition:
  
    F = λ_1 ( x_0 -12 x_1 -3 x_2)^5     +
       λ_2 ( x_0 + 12 x_1 -13 x_2)^5 +
       λ_3 ( x_0 -2 x_1 +3 x_2)^5 +
       λ_4 ( x_0 + 2 x_1 + 3 x_2)^5.

  It remains to compute λ_i's. We can do this easily by solving an
  over-determined linear system, which we know that always has a solution, 
  since the decomposition exists. Doing that, we deduce that 
  λ_1 = 3, λ_2 = 15, λ_3 = 15 and λ_4 = 5.




























































































































§ TENSOR PRODUCT AND SEGRE VARIETIES





 §.§ Introduction: First Approaches

As we saw in the Introduction, if we consider 
the space parametrizing (n_1+1)×⋯× (n_t+1)-tensors
(up to multiplication by scalars), i.e., the space ^N, with
N = ∏_i=1^t (n_i + 1)-1, then additive decomposition problems lead us to study secant varieties of the Segre varieties X_n⊂^N, n=(n_1,…,n_t), which are the image of the Segre embedding of the 
multiprojective spaces ^n_1×⋯×^n_t, defined by the map:

    ν_1,… ,1: ^n_1×⋯×^n_t→^N,

 
    ν_1,… ,1(P) = (a_1,0a_2,0⋯ a_t,0, … ,a_1,n_1⋯ a_t,n_t),

 where
P=((a_1,0,… ,a_1,n_1), …,(a_t,0,… ,a_t,n_t))∈^n_1×⋯×^n_t, and the products are taken in lexicographical order.
 For example, if P= ((a_0,a_1),(b_0,b_1,b_2)) ∈^1×^2, then we have 
 ν_1 ,1(P)= (a_0b_0, a_0b_1, a_0b_2, a_1b_0, a_1b_1,a_1b_2) ∈^5.

Note that, if { x_i,0,… ,x_i,n_i} are homogeneous coordinates in ^n_i and z_j_1,…,j_t, j_i∈{0,… n_i} 
are homogeneous coordinates in ^N, we have
that X_n is the variety whose parametric equations are:

    z_j_1,…,j_t = x_j_1,1⋯ x_j_t,t;      j_i∈{1,… n_i}.
 
Since the use of tensors is ubiquitous in so many applications and
to know a decomposition for a given tensor allows one to ease the computational complexity
when trying to manipulate or study it, this problem has many connections with questions
raised by computer scientists in complexity theory <cit.> and
by biologists and statisticians (e.g., see <cit.>).

As it is to be expected with a problem with so much interest in such
varied disciplines, the approaches have been varied; see, e.g.,
<cit.> for the computational complexity
 for the biological
statistical approach, <cit.> for
the classical algebraic geometry  for the representation theory approach,
<cit.> for a tropical approach and <cit.> for a
multilinear algebra approach. Since the t=2 case is easy (it
corresponds to ordinary matrices), we only consider t≥ 3.

The first fundamental question about these secant varieties, as we
have seen, is about their dimensions. Despite all the progresses made on
this question, it still remains open; only several partial results
are known.

Notice that the case t=3, since it corresponds to the simplest
tensors, which are not matrices, had been widely studied, and many
previous results from several authors are collected in <cit.>.

We start by mentioning the following result on non-degeneracy; see <cit.>.


Let n_1≤ n_2≤⋯≤ n_t, t≥ 3. Then, the dimension of the s-th secant variety of the Segre variety X_ is as expected, i.e.,


    σ_s(X_n) = s(n_1+n_2+⋯+n_t+1)-1


if either:


  * s≤ n_1+1;   or

  * max{ n_t+1,s}≤[ n_1+n_2+⋯+n_t+1 2]. 



In the paper mentioned above, these two results are obtained in two
ways. The first is via combinatorics on monomial
ideals in the multihomogeneous coordinate ring of ^n_1×⋯×^n_t: curiously enough, this corresponds to
understanding possible arrangements of a set of rooks on an
t-dimensional chessboard (corresponding to the array representing
the tensor). There is also a reinterpretation of these problems in
terms of code theory and Hamming distance (the so-called Hamming
codes furnish nice examples of non-defective secants varieties to
Segre's of type ^n×⋯×^n).

Combinatorics turns out to be a nice, but limited tool for those
questions. The second part of Theorem <ref> (and many other results that we are going to report) are obtained by the use of inverse systems and the multigraded version of apolarity theory (recall Section <ref> for the standard case, and we refer to <cit.> for definitions of multigraded apolarity) or via Terracini's lemma (see Lemma <ref>).

The idea behind these methods is to translate the problem of determining the dimension of
σ_s(X), into the problem of determining the
multihomogeneous Hilbert function of a scheme
Z ⊂^n_1×⋯×^n_t of s generic two-fat points in multi-degree 1 = (1,…,1). We have that the coordinate ring of the multi-projective space ^n_1×⋯×^n_t is the polynomial ring S=[x_1,0,… ,x_1,n_1,…,x_t,0,… ,x_t,n_t], equipped with the multi-degree given by (x_i,j) = _i = (0,…,i1,…,0). Then, the scheme Z is defined by a multi-homogeneous ideal I = I(Z), which inherits the multi-graded structure. Hence, recalling the standard definition of Hilbert function (Definition <ref>), we say that the multi-graded Hilbert function of Z in multi-degree = (d_1,…,d_t) is:

    (Z;) = _(S/I)_ = _ S_ - _ I_.






 §.§ The Multiprojective Affine Projective Method

 We describe here a way to study the dimension of σ_s(X_n) by studying the multi-graded Hilbert function of a scheme of fat points in multiprojective space via a very natural reduction to the Hilbert function of fat points in the standard projective space (of equal dimension). 

We start recalling a direct consequence of Terracini's lemma for any variety.

Let Y ⊂^N be a positive dimensional smooth variety, and let Z⊂ Y be a scheme of s generic two-fat points, i.e., a scheme defined by the ideal sheaf ℐ_Z = ℐ^2_P_1,Y∩⋯∩ℐ^2_P_s,Y⊂𝒪_Y, where the P_i's are s generic points of Y defined by the ideal sheaves _P_i,Y⊂_Y, respectively. Since there is a bijection between hyperplanes of the space ^N containing the linear space ⟨ T_P_1(Y),… ,T_P_s(Y) ⟩
and the elements of H^0(Y,ℐ_Z(1)), we have the following consequence of the Terracini lemma.
 

Let Y be a positive dimensional smooth variety; let P_1,…,P_s be generic points on Y; and let Z⊂ Y be the scheme defined by _P_1,Y^2 ∩…∩_P_s,Y^2. Then,

    σ_s(Y) = ⟨ T_P_1(Y),…,T_P_s(Y) ⟩ = N - _
    H^0(Y,ℐ_Z(1)).
 


Now, we apply this result to the case of Segre varieties; we give, e.g., <cit.> as the main reference. Consider ^ := ^n_1×⋯×^n_t, and let X_𝐧⊂^N be its Segre embedding given by 𝒪_^n_1(1)⊗…⊗𝒪_^n_t(1). By applying Theorem <ref> and since the scheme Z ⊂ X_𝐧 corresponds to a scheme of s generic two-fat points in X, which, by a little abuse of notation, we call again Z, we get:

    σ_s(X_𝐧) = (Z, 1)-1.







 
Now, let n=n_1+ ⋯ +n_t, and consider the birational map: 

    ^𝔸^n,

where: 



([x_1,0:…:x_1,n_1],…,[x_t,0,…,x_t,n_t]) 

[origin = c]-90↦ 

(x_1,1 x_1,0,x_1,2 x_1,0,…,x_1,n_1 x_1,0;
 x_2,1 x_2,0,…,x_2,n_2 x_2,0;…; x_t,1
x_t,0,…,x_t,n_t x_t,0).



This map is defined in the open subset of ^ given by {x_1,0x_2,0⋯ x_t,0≠ 0}.

Now, let [z_0,z_1,1,…,z_1,n_1, z_2,1,…,z_2,n_2, … , z_t,1, … ,z_t,n_t] be the coordinate ring of ^n, and consider the embedding of 𝔸^n →^n, whose image is the affine chart { z_0≠ 0 }. By composing the two maps above, we get:

    φ: ^^n,

with: 



([x_1,0:…:x_1,n_1],…,[x_t,0,…,x_t,n_t]) 

[origin = c]-90↦ 

[1:x_1,1 x_1,0:x_1,2 x_1,0:…:x_1,n_1 x_1,0:
 x_2,1 x_2,0:…:x_2,n_2 x_2,0:…:x_t,1
x_t,0:…:x_t,n_t x_t,0].



Let Z⊂ X be a zero-dimensional scheme, which is contained in the affine chart {x_0,1x_0,2⋯ x_0,t≠ 0}, and let Z' = φ(Z). We want to construct 
a scheme W⊂^n such that (W;d) = (Z;(d_1,…,d_t)), where d= d_1+… + d_t.

Let:

    Q_0,Q_1,1,…,Q_1,n_1,Q_2,1,…,Q_2,n_2,Q_t,1,…,Q_t,n_t

be the coordinate points of ^n. Consider the linear space Π_i≅^n_i-1⊂^n, where Π_i=⟨ Q_i,1,…,Q_i,n_i⟩. The defining ideal of Π_i is:

    I(Π_i) = (z_0, z_1,1, … , z_1,n_1; … ; ẑ_i,1,… ,ẑ_i,n_i; 
    … ;z_t,1, … , z_t,n_t)  .

Let W_i be the subscheme of ^n denoted by (d_i-1)Π_i, i.e., the scheme defined by the ideal I(Π_i)^d_i-1. Since I(Π_i) is a prime ideal generated by a regular sequence, the ideal I(Π_i)^d_i-1 is saturated (and even primary for I(Π_i)). 
Notice that W_i∩ W_j = ∅, for i≠ j. With this construction, we have the following key result.


Let Z, Z', W_1, … ,W_t be as above, and let
W = Z' + W_1 + ⋯ + W_t⊂^n. Let I(W) ⊂ S and I(Z) ⊂ R be the ideals of 
W and Z, respectively. Then, we have, for all (d_1,… ,d_t)∈ℕ^t: 

    _ I(W)_d = _ I(Z)_(d_1,… ,d_t) ,

where d = d_1 + … + d_t.



Note that when studying Segre varieties, we are only interested to the case (d_1,… ,d_t) = (1, … , 1); but, in the more general case of Segre–Veronese varieties, we will have to look at Theorem <ref> for any multidegree (d_1,…,d_t); see Section <ref>.

Note that the scheme W in ^n that we have constructed has two parts: the part W_1 + ⋯ + W_t (which we shall call the part at infinity and we denote as W_∞) and 
the part Z^', which is isomorphic to our original zero-dimensional
scheme Z ⊂ X. Thus, if Z = ∅ (and hence, Z^' = ∅), we obtain from the theorem that:

    _ I(W_∞)_d = _ S_(d_1, … , d_t),    d = d_1 + ⋯ + d_t.

It follows that:

    (W_∞;d) = d_1 + ⋯ + d_t + n n - d_1+n_1 n_1⋯d_t+n_t n_t.

With this observation made, the following corollary is immediate.

 
Let Z and Z^' be as above, and write W = Z^' + W_∞. 
Then,

    (W;d) = (Z;(d_1,… ,d_t)) + (W_∞;d).

 
Eventually, when Z is given by s generic two-fat points in multi-projective space, we get the following.

Let Z⊂ X be a generic set of s two-fat points, and let W⊂^n be as in the Theorem <ref>. Then, we have: 

    σ_s( X_𝐧) = (Z,(1,…,1))-1 = N -  I(W)_t,

where N = Π_i=1^t(1 + n_i) - 1.


Therefore, eventually, we can study a projective scheme W ⊂^ n, which is made of a union of generic two-fat points and of fat linear spaces. Note that, when n_1=…=n_t=1, then also W is a scheme of fat points. 





 §.§ The Balanced Case
 One could try to attack the problem starting
with a case, which is in a sense more “regular”, i.e., the “balanced”
case of n_1=⋯ =n_t, t≥ 3. Several partial results are
known, and they lead Abo, Ottaviani and Peterson to propose, in
their lovely paper <cit.>, a conjecture, which states that
there are only a finite number of defective Segre varieties of the
form ^n×⋯×^n, and their guess is that
σ_4(^2×^2×^2) and
σ_3(^1×^1×^1×^1) are
actually the only defective cases (as we will see later, this is just
part of an even more hazardous conjecture; see Conjecture <ref>).

In the particular case of n_1=⋯ =n_t= 1, the question has been completely solved in
<cit.>, supporting the above conjecture.

<cit.>
Let t, s ∈ N, t ≥ 3. Let X_1 be the Segre embedding of ^1×⋯×^1, (t-times). The dimension of σ_s(X_1) ⊂^N, with N = 2^t -1, is always as expected, i.e., 
    σ_s(X) = min{ N, s(t+1)-1 },
 except for t = 4, s=3. In this last case,
 σ_3(X) = 13, instead of 14.

The method that has been used to compute the multi-graded
Hilbert function for schemes of two-fat points with generic support in multi-projective spaces
is based first on the procedure described on the multiprojective affine projective method explained in the previous section, which brings to the study of the standard Hilbert function of the schemes W⊂^t. Secondly, the problem of determining the dimension of I(W)_t can be attacked by induction, via the powerful tool constituted by the differential Horace method, created by Alexander and Hirschowitz; see Section <ref>. This is used in <cit.> together with other “tricks”, which allow one to “move” on a hyperplane some of the conditions imposed by the fat points, analogously as we have described in the examples in Section <ref>. These were the key ingredients to prove Theorem <ref>.

The only defective secant variety in the theorem above is made by
the second secant variety of ^1×^1×^1×^1
⊂^15, which, instead of forming a hypersurface in
^15, has codimension two. Via Theorem <ref> above, this is geometrically related to a configuration of seven fat points; more precisely, in this case, the scheme W of Theorem <ref> is union of three two-fat points and four three-fat ones (see also Theorem <ref> for a detailed description of W). These always lie on a rational normal curve in ^4 (see, e.g., Theorem 1.18 of <cit.>) and do not have the expected Hilbert function in degree four, by the result in <cit.>.

For the general “balanced” case ^n×⋯×^n, the
following partial result is proven in <cit.>.

 
Let X_ be the Segre embedding of ^n×⋯×^n (t times), t≥ 3. Let s_t and e_t be defined by:

    s_t=⌊(n+1)^t/nt+1⌋ 8pt and 8pt e_t≡ s_tmod (n+1)  8pt  with 8pt e_t∈{0,… ,n}.

Then:


  * if s≤ s_t-e_t, then σ_s(X) has the expected
dimension;

  * if s≥ s_t-e_t+n+1, then σ_s(X) fills the ambient
space.



In other words, if s_t =q (n+1)+r, with 1 ≤ r ≤ n, then σ_s(X_) has the expected dimension both for s ≥ (q+1)(n+1) and for s ≤ q(n+1), but if n+1 divides s_t, then σ_s(X) has the expected dimension for any s.


Other known results in the “balanced” case are the following:



  * σ_4(^2×^2×^2) is defective with
defect δ_4=1; see <cit.>.


  * ^n ×^n×^n is never defective for n ≥
3; see <cit.>; 


  * ^n ×^n×^n×^n for 2 ≤ n ≤ 10 is never defective except at most for s = 100 and n = 8 or for s = 357
and n = 10; see <cit.>. 





 §.§ The General Case

When we drop the balanced dimensions request, not many defective
cases are actually known. For example:


  * ^2 ×^3×^3 is defective only for s=5, with defect δ_4=1; see <cit.>;

  * ^2 ×^n ×^n with n even is defective
only for s=3n/2+1 with defect δ_s=1; see <cit.>.

  * ^1 ×^1×^n×^n is defective only
for s=2n+1 with defect δ_2n+1=1; see <cit.>.


However, when taking into consideration cases where the n_i's are far
from being equal, we run into another “defectivity phenomenon”, known as
“the unbalanced case”; see <cit.>.

 
Let X_ be the Segre embedding of P^n_1×⋯× P^n_t× P^n ⊂ ^M, with M=(n+1)Π _i=1 ^t ( n_i +1)-1.
Let N=Π _i=1 ^t ( n_i +1)-1, and assume n > N - ∑ _i=1^tn_i + 1. Then, σ_s(X) is defective for:

    N - ∑ _i=1 ^t n_i +1 < s ≤min{ n;N},

with defect equal to δ_s (X) = s^2-s(N-∑_i=1^tn_i +1).


The examples described above are the few ones for which
defectivities of Segre varieties are known. Therefore, the following
conjecture has been stated in <cit.>, where, for n_1≤ n_2 ≤…≤ n_t, it is proven for s ≤ 6.


The Segre embeddings of ^n_1×⋯. ×^n_t,
t≥ 3, are never defective, except for:


  * ^1 ×^1 ×^1×^1, for s=3, with
δ_3=1;

  * ^2 ×^2 ×^2, for s=4, with
δ_4=1;

  * the “unbalanced case”;

  * ^2 ×^3 ×^3, for s=5, with
δ_5=1;

  * ^2 ×^n ×^n, with n even, for s=3n/2+1, with
 δ_s=1);

  * ^1 ×^1 ×^n ×^n, for s=2n+1, with δ_2n+1=1).











§ OTHER STRUCTURED TENSORS


There are other varieties of interest, parametrizing other “structured tensors”, i.e., tensors that have determined properties. In all these cases, there exists an additive decomposition problem, which can be geometrically studied similarly as we did in the previous sections. In this section, we want to present some of these cases. 

In particular, we consider the following structured tensors:

	
  * skew-symmetric tensors, i.e., v_1∧…∧ v_k ∈⋀^k V;
	
  * decomposable partially-symmetric tensors, i.e., L_1^d_1⊗…⊗ L_t^d_t∈ S^d_1V_1^*⊗⋯⊗ S^d_tV_t^*;
	
  * d-powers of linear forms, i.e., homogeneous polynomials L_1^d_1⋯ L_t^d_t∈ S^d V^*, for any partition d = (d_1,…,d_t) ⊢ d;
	
  * reducible forms, i.e., F_1⋯ F_t ∈ S^dV^*, where (F_i) = d_i, for any partition = (d_1,…,d_t) ⊢ d;
	
  * powers of homogeneous polynomials, i.e., G^k ∈ S^d V^*, for any k | d;





 §.§ Exterior Powers and Grassmannians


Denote by (k,n) the Grassmannian of k-dimensional linear subspaces of
^n ≅ V, for a fixed n+1 dimensional vector space
V. We consider it with the embedding given by its Plücker coordinates as embedded in ^N_k, where N_k = n+1 k -1.

The dimensions of the higher secant varieties to the Grassmannians
of lines, i.e., for k=1, are well known; e.g., see <cit.>,
<cit.> or <cit.>. The secant variety σ_s(Gr(1,n))
parametrizes all (n+1)× (n+1) skew-symmetric matrices of rank at most 2s.

 
We have that σ_s((1,n)) is
defective for s < ⌊n+1  2⌋ with defect equal to δ_s =
2s(s-1).


For k≥ 2, not many results can be found in the classical or
contemporary literature about this problem; e.g., see <cit.>. However, they
are sufficient to have a picture of the whole situation.
Namely, there are only four other cases that are known to be
defective (e.g., see <cit.>), and it is conjectured in
<cit.> that these are the only ones. This is summarized in the following conjecture:

 
Let k≥ 2. Then, the secant variety σ_s((k,n)) has the expected dimension
except for the following cases:
 
    [                        actual codimension expected codimension;           σ_3((2,6))                    1                    0;           σ_3((3,7))                   20                   19;           σ_4((3,7))                    6                    2;           σ_4((2,8))                   10                    8;                      ]



In <cit.>, they proved the conjecture for n≤ 15 (the case n≤ 14 can be found in <cit.>). The conjecture has been proven to hold for s≤ 6 (see
<cit.>) and later for s≤ 12 in <cit.>.

A few more results on non-defectivity are proven in
<cit.>. We summarize them
in the following.

 The secant variety σ_s((k,n)) has the
expected dimension when:


  * k≥ 3 and ks≤ n+1,


  * k=2, n≥ 9 and s≤ s_1(n) or s≥ s_2(n), 

where:


    s_1(n)=⌊n^2/18-20n/27+287/81⌋+
    ⌊6n-13/9⌋ and  
     s_2(n)=⌈n^2/18-11n/27+44/81⌉+
    ⌈6n-13/9⌉;


in the second case, σ_s((2,n)) fills the ambient space.

Other partial results can be found in <cit.>, while in <cit.>, the following theorem can be found.


Given an integer m ≥ 2, we define a function
h_m : ℕ→ℕ
as follows: 


  * h_m(0) = 0;

  * for any k ≥ 1, write
k + 1 = 2^λ_1 + 2^λ_2 + … + 2^λ_ℓ + ϵ, for a suitable choice of
λ_1 > λ_2 > … > λ_ℓ≥ 1, ϵ∈{0, 1}, and define:

    h_m(k) := m^λ_1-1 + m^λ_2-1 + … + m^λ_ℓ-1.

In particular, we get h_m(2k) = h_m(2k - 1), and h_2(k) = ⌊k+1/2⌋.



Assume that k ≥ 2, and set:

    α := ⌊n+1/k+1⌋.

If either:


  * n ≥ k^2 + 3k + 1 and h ≤α h_α(k - 1);

or:


  * n < k^2 + 3k + 1, k is even, and h ≤ (α - 1)h_α(k - 1) + h_α(n - 2 - α r);

or:


  * n < k^2+3k+1, r is odd, and: 
    h ≤ (α - 1)h_α(k-2)+h_α(min{n-3-α (k-1), r-2}),

 
then, (k, n) is not (h + 1)-defective.



This results strictly improves the results in <cit.> for k ≥ 4, whenever (k, n)≠ (4, 10), (5,11); see <cit.>.

Notice that, if we let _∧(k,n) be the generic rank with respect to (k,n), i.e., the minimum s such that σ_s((k,n))=^N_k, we have that the results above give
that, asymptotically: 

    _∧(2,n) ∼n^2  18.

A better asymptotic result can be found in <cit.>.


 Finally, we give some words concerning the methods involved.
The approach in <cit.> uses Terracini's lemma and an
exterior algebra version of apolarity. The main idea there is to consider the analog of the perfect pairing induced by the apolarity action that we have seen for the symmetric case in the skew-symmetric situation; see Section <ref>. In fact, the pairing considered here is: 
    ⋀^kV ×⋀^n-kV →⋀^n V ≃,
 induced by the multiplication in ⋀ V, and it defines the apolarity of a subspace Y⊂ V of dimension k to be Y^⊥:={ w ∈⋀^n-kV   |   w∧ v =0  ∀  v ∈ Y}. Now, one can proceed in the same way as the symmetric case, namely by considering a generic element of the Grassmannian (k,n) and by computing the tangent space at that point. Then, its orthogonal, via the above perfect pairing, turns out to be, as in the symmetric case, the degree n-k part of an ideal, which is a double fat point. Hence, in <cit.>, the authors apply Terracini's lemma to this situation in order to study all the known defective cases and in other various cases. Notice that the above definition of skew-symmetric apolarity works well for computing the dimension of secant varieties to Grassmannians since it defines the apolar of a subspace that is exactly what is needed for Terracini's lemma, but if one would like to have an analogous definition of apolarity for skew symmetric tensors, then there are a few things that have to be done. Firstly, one needs to extend by linearity the above definition to all the elements of ⋀^kV. Secondly, in order to get the equivalent notion of the apolar ideal in the skew symmetric setting, one has to define the skew-symmetric apolarity in any degree ≤ d. This is done in <cit.>, where also the skew-symmetric version of the apolarity lemma is given. Moreover, in <cit.>, one can find the classification of all the skew-symmetric-ranks of any skew-symmetric tensor in ⋀^3ℂ^n for n≤ n (the same classification can actually be found also in <cit.>), together with algorithms to get the skew-symmetric-rank and the skew-symmetric decompositions for any for those tensors (as far as we know, this is new).

Back to the results on dimensions of secant varieties of Grassmannians: in <cit.>, a tropical geometry approach is involved. In <cit.>, as was done by Alexander and Hirshowitz for the symmetric case, the authors needed to introduce a specialization technique, by placing a certain number of
points on sub-Grassmannians and by using induction. In this way, they could prove several non-defective cases. Moreover, in the same work, invariant theory was used to describe the equation of σ_3((2,6)), confirming the work of Schouten <cit.>, who firstly proved that it was defective by showing that it is a hypersurface (note that by parameter count, it is expected to fill the ambient space). Lascoux <cit.> proved that the degree of Schouten's hypersurface is seven. In <cit.>, with a very clever idea, an explicit description of this degree seven invariant was found by relating its cube to the determinant of a 21 × 21 symmetric matrix.

Eventually, in <cit.>, the author employed a new method for studying the defectivity of varieties based on the study of osculating spaces. 






 §.§ Segre–Veronese Varieties

Now, we consider a generalization of the apolarity action that we have seen in both Section <ref> and Section <ref> to the multi-homogeneous setting; see <cit.>. More precisely, fixing a set of vector spaces V_1,…,V_t of dimensions n_1+1,…,n_t+1, respectively, and positive integers d_1,…,d_t, we consider the space of partially-symmetric tensors: 
    S^d_1V^*_1⊗…⊗ S^d_tV^*_t.

 The Segre–Veronese variety parametrizes decomposable partially-symmetric tensors, i.e., it is the image of the embedding:

    [                       ν_, :  ℙ(S^1V_1^*) ×…×ℙ(S^1V_t^*)                           ⟶ ℙ(S^d_1V_1^*⊗…⊗ S^d_tV_t^*);                                         ([L_1],…,[L_t])                           ↦       [L_1^d_1⊗…⊗ L_t^d_t], ]

where, for short, we denote = (n_1,…,n_t),  = (d_1,…,d_t).

More geometrically, the Segre–Veronese variety is the image of the Segre–Veronese embedding: 

    ν_, : ℙ^ := ℙ^n_1×…×ℙ^n_t⟶ℙ^N,    where  N = ∏_i=1^t d_i+n_i  n_i - 1,

given by 𝒪_ℙ^() := 𝒪_ℙ^n_1(d_1)⊗…⊗𝒪_ℙ^n_t(d_t), that is via the forms of 
 multidegree (d_1,… ,d_t) of the multigraded homogeneous coordinate ring: 

    R = [x_1,0,… ,x_1,n_1;x_2,0,… ,x_2,n_2;… ;x_t,0,… ,x_t,n_t].

For instance, if = (2,1), =(1,2) and P= ([a_0:a_1:a_2],[b_0:b_1]) ∈^2×^1, we have 
 ν_,(P)=
[a_0b_0^2:a_0b_0b_1:a_0b_1^2:a_1b_0^2: a_1b_0b_1: a_1b_1^2: a_2b_0^2: a_2b_0b_1: a_2b_1^2], where the products are taken in lexicographical order.
We denote the embedded variety ν_,(ℙ^) by X_,. Clearly, for t=1, we recover Veronese varieties, while for (d_1,… ,d_t)= (1,…,1), we get the Segre varieties.


The corresponding additive decomposition problem is as follows.

	Given a partially-symmetric tensor T ∈ S^d_1 V_1^* ×…×ℙS^d_t V_t^* or, equivalently, a multihomogeneous polynomial T ∈ R of multidegree , find the smallest possible length r of an expression T = ∑_i=1^r L_i,1^d_1⊗…⊗ L_i,t^d_t.


As regards the generic tensor, a possible approach to this problem mimics what has been done for
Segre and Veronese varieties. One can use Terracini's lemma (Lemma <ref> and Theorem <ref>), as in <cit.>, to translate the problem of determining the dimensions of the higher secant varieties of X_, into that of calculating the value, at =(d_1,… ,d_t), of the Hilbert function of generic sets of two-fat points in ℙ^.
Then, by using the multiprojective affine projective method introduced in Section <ref>, i.e., by passing to an affine chart in ℙ^
and then homogenizing in order to pass to ℙ^n, with
n=n_1+⋯+n_t, this last calculation amounts to computing the
Hilbert function in degree d = d_1 + ⋯ + d_n for the
subscheme W⊂ℙ^n; see Theorem <ref>.

There are many scattered results on the dimension of
σ_s(X_,), by many authors, and
very few general results. One is the following, which generalizes
the “unbalanced” case considered for Segre varieties; see
<cit.>.

  Let X = X_(n_1, … , n_t, n),(d_1, … , d_t,1) be the
Segre–Veronese embedding:

    ^n_1×…×^n_t×^n (d_1,…,d_t,1)⟶ X
     ⊂^M,    with  M =(n+1)(∏_i=1^tn_i + d_i d_i) - 1.

Let N = Π_i=1^tn_i + d_i d_i - 1, then, for
N-∑_i=1^tn_i +1 < s ≤min{n,N}, the secant variety
 σ_s (X) is defective with δ_s (X) =
s^2-s(N-∑_i=1^tn_i +1).


When it comes to Segre–Veronese varieties with only two factors, there are
many results by many authors, which allow us to get a quite complete
picture, described by the following conjectures, as stated in
<cit.>.

 Let X = X_(m,n),(a,b), then X is never defective, except for:


  * b=1, m≥2, and it is unbalanced (as in the theorem above); 
  * (m,n)= (1,n), (a,b)=(2d,2);

  * (m,n)= (3,4), (a,b)=(2,1);

  * (m,n)= (2,n), (a,b)=(2,2);

  * (m,n)= (2,2k+1), k≥ 1, (a,b)=(1,2);

  * (m,n)= (1,2), (a,b)=(1.3);

  * (m,n)= (2,2), (a,b)=(2,2);

  * (m,n)= (3,3), (a,b)=(2,2);

  * (m,n)= (3,4), (a,b)=(2,2).



 Let X = X_(m,n),(a,b), then for (a,b) ≥ (3,3), X is never defective.




The above conjectures are based on results that can be found in <cit.>.


For the Segre embeddings of many copies of P^1, we have a complete result. First, in <cit.> and in <cit.>, the cases of two and three copies of P^1, respectively, were completely solved.


Let X = X_(1,1),(a,b), a≤ b. Then, X is never defective, except for
(a,b) = (2,2d); in this case, σ_2d+1(X) is defective
with δ_2d+1=1.



Let X = X_(1,1,1),(a,b,c), a≤ b ≤ c; then X is never defective, except for:


  * (a,b,c) = (1,1,2d); in this case, σ_2d+1(X) is defective
with δ_2d+1=1;

  * (a,b,c) =(2,2,2); here, σ_7(X) is defective, and
δ_7=1.


In <cit.> the authors, by using an induction approach, whose basic step was Theorem <ref> about the Segre varieties X_1 <cit.>, concluded that there are no other defective cases except for the previously-known ones.


Let X = X_(1,…,1),(a_1,… ,a_r). Then, X is never defective, except for:


  * X_(1,1),(2,2d) (Theorem <ref>);

  * X_(1,1,1),(1,1,2d) and X_(1,1,1),(2,2,2) (Theorem <ref>);

  * X_(1,1,1,1),(1,1,1,1) (Theorem <ref>).



For several other partial results on the defectivity of certain Segre–Veronese varieties, see, e.g., <cit.>, and for an asymptotic result about non-defective Segre–Veronese varieties, see <cit.>.




 §.§ Tangential and Osculating Varieties to Veronese Varieties

Another way of generalizing what we saw in Section <ref> for secants of Veronese Varieties X_n,d is to work with their tangential and osculating varieties. 

 
Let X_n,d⊂ ^N be a Veronese variety. We denote by
 τ (X_n,d) the tangential variety of X_n,d, i.e., the closure in ^N of the union of all tangent spaces: 
 
    τ(X_n,d) = ⋃ _P∈ X_n,d T_P(X_n,d)⊂^N.

 More in general, we denote by O^k(X_n.d) the k-th osculating variety of X_n,d, i.e., the closure in ^N of the union of all k-th osculating spaces:
 
    O^k(X_n,d) = ⋃ _P∈ X_n,d
    O^k_P(X_n,d)⊂^N.

Hence, τ (X_n,d) = O^1(X_n,d).

These varieties are of interest also because the space O^k(X_n,d) parametrizes a particular kind of form. Indeed, if the point P = [L^d] ∈ X_n,d⊂ S_d, then the k-th osculating space O^k_P(X_n,d) correspond to linear space {[L^d-kG]  |  G ∈ S_k}. Therefore, the corresponding additive decomposition problem asks the following.

	Given a homogeneous polynomial F ∈[x_0,…,x_n], find the smallest length of an expression F = ∑_i=1^r L_i^d-kG_i, where the L_i's are linear forms and the G_i's are forms of degree k.

The type of decompositions mentioned in the latter problem have been called generalized additive decomposition in <cit.> and in <cit.>. In the special case of k = 1, they are a particular case of the so-called Chow–Waring decompositions that we treat in full generality in Section <ref>. In this case, the answer to Problem <ref> is called (d-1,1)-rank, and we denote it by _(d-1,1)(F).

	Given a family of homogeneous polynomials = {F_1,…,F_m}, we define the simultaneous rank of  the smallest number of linear forms that can be used to write a Waring decomposition of all polynomials of . 
	
	Now, a homogeneous polynomial F ∈ S^dV can be seen as a partially-symmetric tensor in S^1V ⊗ S^d-1V via the equality: 
	
    F = 1/d∑_i=0^n x_i ⊗∂ F/∂ x_i.

	From this expression, it is clear that a list of linear forms that decompose simultaneously all partial derivatives of F also decompose F, i.e., the simultaneous rank of the first partial derivatives is an upper bound of the symmetric-rank of F. Actually, it is possible to prove that for every homogeneous polynomial, this is an equality (e.g., see <cit.> (Section 1.3) or <cit.> (Lemma 2.4)). For more details on relations between simultaneous ranks of higher order partial derivatives and partially-symmetric-ranks, we refer to <cit.>.

Once again, in order to answer the latter question in the case of the generic polynomial, we study the secant varieties to the k-th osculating variety of X_n,d. In <cit.>, the dimension of σ_s(τ (X_n,d)) is studied (k=1 case). Via apolarity and inverse systems, with an analog of Theorem <ref>, the problem is again reduced to the computation of the Hilbert function of some particular zero-dimensional subschemes of ^n; namely,

    σ_s (τ(X_n,d))     = _
    (L_1^d-1,…,L_s^d-1,L_1^d-2M_1,…,L_s^d-2M_s)_d -1 = 
        = (Z,d)-1,

where L_1,…,L_s,M_1,…,M_s are 2s generic linear forms in
[x_0,…,x_n], while (Z,d) is the Hilbert function of a scheme
Z, which is the union of s generic (2,3)-points in ^n, which are defined as follows.

A (2,3)-point is a zero-dimensional scheme in ^n with support
at a point P and whose ideal is of type I(P)^3 +
I(ℓ)^2, where I(P) is the homogeneous ideal of P and ℓ⊂^n is a line through P defining ideal I(ℓ). 

Note that when we say that Z is a scheme of s generic (2,3)-points in ^n, we mean that I(Z) = I(Q_1) ∩…∩ I(Q_s), where the Q_i's are (2,3)-points, i.e., I(Q_i) = I(P_i)^3 + I(ℓ_i)^2, such that P_1,… P_s are generic points in ^n, while ℓ_1,…,ℓ_s are generic lines passing though P_1,…,P_s, respectively.

By using the above fact, in <cit.>, several cases where σ_s (τ(X_n,d)) is defective were found, and it was conjectured that these exceptions were the only ones. The conjecture has been proven in a few cases in <cit.> (s≤ 5 and n ≥ s+1) and in <cit.> (n = 2,3). In <cit.>, it was proven for n ≤ 9, and moreover, it was proven that if the conjecture holds for d = 3, then it holds in every case. Finally, by using this latter fact, Abo and Vannieuwenhoven completed the proof of the following theorem <cit.>.

The s-th secant variety σ_s(τ(X_n,d)) of the tangential variety to the Veronese variety has dimension as expected, except in the following cases:


  * d = 2 and 2 ≤ 2s < n;

  * d = 3 and n = 2,3,4.


As a direct corollary of the latter result, we obtain the following answer to Problem <ref> in the case of generic forms.

	Let F∈ S_d be a generic form. Then,
	
    _(d-1,1)(F) = ⌈n+d  n/2n+1⌉,

	except for: 
	
		
  * d = 2, where _(d-1,1)(F) = ⌊n/2⌋ + 1;
		
  * d = 3 and n = 2,3,4, where _(d-1,1)(F) = ⌈n+d  n/2n+1⌉ + 1.
	


The general case of σ_s(O^k(X_n.d)) is studied in
<cit.>. Working in
analogy with the case k=1, the dimension of
σ_s(O^k(X_n,d) is related to the Hilbert function of a
certain zero-dimensional scheme Z = Z_1∪⋯∪ Z_s, whose support is a generic
union of points P_1,…,P_s∈ ^n, respectively, and such that, for each i = 1,…,s, we have that (k+1)P_i ⊂ Z_i ⊂ (k+2)P_i.

As one of the manifestations of the ubiquity of fat points, the following conjecture describes the conditions for the defect of this secant variety in terms of the Hilbert function of fat points:


 The secant variety σ_s(O^k(X_n,d)) is defective if and only if either:


  * h^1(ℐ_X(d)) > max{0,  Z - d+n 
n}, or

  * h^0(ℐ_T(d)) > max{0, d+n  n - 
Z},
 
where X is a generic union of s (k+1)-fat points and T is a generic union of (k+2)-fat points.


In <cit.>, the conjecture is proven for n=2, s≤ 9, and in <cit.> for n=2 and any s.





 §.§ Chow–Veronese Varieties

Let = (d_1,…,d_t) be a partition of a positive integer d, i.e., d_1 ≥…≥ d_t are positive integers, which sum to d. Then, we consider the following problem.

Let S = ⊕_d ∈ S_d be a polynomial ring in n+1 variables.

	Given a homogeneous polynomial F ∈ S_d, find the smallest length of an expression F = ∑_i=1^r L_i,1^d_1⋯ L_i,t^d_t, where L_i,j's are linear forms.

The decompositions considered in the latter question are often referred to as Chow–Waring decompositions. We call the answer to Problem <ref> as the -rank of F, and we denote it by _(F).

In this case, the summands are parametrized by the so-called Chow–Veronese variety, which is given by the image of the embedding:

    [                ν_ :      ℙ S_1 ×…×ℙ S_1                   ⟶                S_d,;                         ([L_1],…,[L_t])                   ↦ [L_1^d_1⋯ L_t^d_t]. ]

We denote by X_ the image ν_(^). Notice that this map can be seen as a linear projection of the Segre–Veronese variety X_,⊂(S_d_1⊗…⊗ S_d_t), for = (n,…,n), under the map induced by the linear projection of the space of partially-symmetric tensors S_d_1⊗…⊗ S_d_t on the totally symmetric component S_d. Once again, we focus on the question posed in Problem <ref> in the case of a generic polynomial, for which we study dimensions of secant varieties to X_.

In the case of = (d-1,1), we have that X_ coincides with the tangential variety of the Veronese variety τ(X_n,d), for which the problem has been completely solved, as we have seen in the previous section (Theorem <ref>).

The other special case is given by = (1,…,1), for d ≥ 3. In this case, X_d has been also referred to as the Chow variety or as the variety of split forms or completely decomposable forms. After the first work by Arrondo and Bernardi <cit.>, Shin found the dimension of the second secant variety in the ternary case (n=2) <cit.>, and Abo determined the dimensions of higher secant varieties <cit.>. All these cases are non-defective. It is conjectured that varieties of split forms of degree d ≥ 3 are never defective. New cases have been recently proven in <cit.>.

The problem for any arbitrary partition  has been considered in <cit.>. Dimensions of all s-th secant varieties for any partition haves been computed in the case of binary forms (n=1). In a higher number of variables, the dimensions of secant line varieties (s=2) and of higher secant varieties with s ≤ 2 ⌊n/3⌋ have been computed. This was done by using the classical Terracini's lemma (Lemma <ref>) in order to obtain a nice description of the generic tangent space of the s-th secant variety. In the following example, we explain how the binary case could be treated.
 

	If P = [L_1^d_1⋯ L_t^d_t] ∈ X_, then it is not difficult to prove (see Proposition 2.2 in <cit.>) that: 
	
    T_P X_ = ((I_P)_d),    with d = d_1+…+d_t

	where I_P = (L_1^d_1-1L_2^d_2⋯ L_t^d_t,L_1^d_1L_2^d_2-1⋯ L_t^d_t,…,L_1^d_1L_2^d_2⋯ L_t^d_t-1).
	
	In the particular case of binary forms, some more computations show that actually,
	T_P X_ = ((I'_P)_d), where I'_P is the principal ideal (L_1^d_1-1⋯ L_t^d_t-1). In this way, by using Terracini's lemma, we obtain that, if Q is a generic point on the linear span of s generic points on X_, then:
	
    T_Qσ_s(X_) = ((L_1,1^d_1-1⋯ L_1,t^d_t-1,…,L_s,1^d_1-1⋯ L_s,t^d_t-1)_d),

	where L_i,j's are generic linear forms. Now, in order to compute the dimension of this tangent space, we can study the Hilbert function of the ideal on the right-hand side. By semicontinuity, we may specialize to the case L_i,1 = … = L_i,t, for any i = 1,…,s. In this way, we obtain a power ideal, i.e., an ideal generated by powers of linear forms, whose Hilbert function is prescribed by Fröberg–Iarrobino's conjecture; see Remark <ref>. Now, since in <cit.>, the authors proved that the latter conjecture holds in the case of binary forms, i.e., the Hilbert function of a generic power ideal in two variables is equal to the right-hand side of (<ref>), we can conclude our computation of the dimension of the secant variety of X_ in the binary case. This is the way Theorem 3.1 in <cit.> was proven.


In the following table, we resume the current state-of-the-art regarding secant varieties and Chow–Veronese varieties.





	     s     d     n     References     σ_s(X_) 

	(d-1,1)     any     any     any     <cit.>     non-defective, except for 
	
	                    (1) d = 2 and 2 ≤ 2s < n;

	                    (2) d = 3 and n = 2,3,4. 

	(1,…,1)        d>2     3(s-1)<n     <cit.>     non-defective

	     any     any     2     <cit.>     non-defective, except for cases above

	      3c | some numerical constraints     <cit.>      

	any     any     any     1     <cit.>     non-defective, except for cases above 

	    2     any     any         

	    ≤ 2 ⌊n/3⌋     any     any     







 §.§ Varieties of Reducible Forms

In 1954, Mammana <cit.> considered the variety of reducible plane curves and tried to generalize previous works by, among many others, C. Segre, Spampinato and Bordiga. More recently, in <cit.>, the authors considered the varieties of reducible forms in full generality. 

Let = (d_1,…,d_t) ⊢ d be a partition of a positive integer d, i.e., d_1 ≥…≥ d_t are positive integers, which sum up to d and t ≥ 2. Inside the space of homogeneous polynomials of degree d, we define the variety of -reducible forms as:

    Y_ = {[F] ∈ S_d  |  F = G_1⋯ G_t,  where (G_i) = d_i},

i.e., 
the image of the embedding:

    [              ψ_ : ℙ S_d_1×…×ℙ S_d_t                 ⟶              S_d,;                         (G_1,…,G_t)                 ↦ G_1^d_1⋯ G_t^d_t. ]

Clearly, if d = 2, then = (1,1), and Y_(1,1) is just the Chow variety X_(1,1). In general, we may see Y_ as the linear projection of the Segre variety X_m inside (S_d_1⊗⋯⊗ S_d_t), where m = (m_1,…,m_t) with m_i = d_i+n  n - 1. Note that, if , ' are two partitions of d such that  can be recovered from ' by grouping and summing some of entries, then we have the obvious inclusion Y_'⊂ Y_. Therefore, if we define the variety of reducible forms as the union over all possible partitions ⊢ d of the varieties Y_, we can actually write:

    Y = ⋃_k = 1^⌊d/2⌋ Y_(d-k,k)⊂ S_d.

In terms of additive decompositions, the study of varieties of reduced forms and their secant varieties is related to the notion of the strength of a polynomial, which was recently introduced by T. Ananyan and M. Hochster <cit.> and then generalized to any tensor in <cit.>. 

	Given a homogeneous polynomial F ∈ S_d, find the smallest length of an expression F = ∑_i=1^r G_i,1G_i,2, where 1 ≤(G_i,j) ≤ d-1.

The answer to Problem <ref> is called the strength of F, and we denote it by S(F). 

In <cit.>, the authors gave a conjectural formula for the dimensions of all secant varieties σ_s(Y_) of the variety of -reducible forms for any partition  (see Conjecture 1.1 in <cit.>), and they proved it under certain numerical conditions (see Theorem 1.2 in <cit.>). These computations have been made by using the classical Terracini's lemma and relating the dimensions of these secants to the famous Fröberg's conjecture on the Hilbert series of generic forms. 

The variety of reducible forms is not irreducible and the irreducible component with biggest dimension is the one corresponding to the partition (d-1,1), i.e., Y =  Y_(d-1,1). Higher secant varieties of the variety of reducible forms are still reduced, but understanding which is the irreducible component with the biggest dimension is not an easy task. In Theorem 1.5 of <cit.>, the authors proved that, if 2s ≤ n-1, then the biggest irreducible component of σ_s(Y) is σ_s(Y_(d-1,1), i.e., σ_s(Y) = σ_s(Y_(d-1,1)), and together with the aforementioned Theorem 1.2 of <cit.>, this allows us to compute the dimensions of secant varieties of varieties of reducible forms and answer Problem <ref> under certain numerical restrictions (see Theorem 7.4 <cit.>). 

In conclusion, we have that Problem <ref> is answered in the following cases:

	
  * any binary form (n=2), where S(F) = 1,
	
	 since every binary form is a product of linear forms;
	
  * generic quadric (d=2), where S(F) = ⌊n/2⌋ +1, 
	
	since it forces = (1,1), which is solved by Corollary <ref> (2);
	
  * generic ternary cubic (n = 2, d=3), where S(F) = 2, 
	
	since Y_(2,1) is seven-dimensional and non-degenerate inside ^9 =  S_3, then σ_2(Y_(2,1)) cannot be eight-dimensional; otherwise, we get a contradiction by one of the classical Palatini's lemmas, which states that if σ_s+1(X) = σ_s(X) + 1, then σ_s+1(X) must be a linear space <cit.>.




 §.§ Varieties of Powers

Another possible generalization of the classical Waring problem for forms is given by the following.

	Given a homogeneous polynomial F ∈ S_d and a positive divisor k > 1 of d, find the smallest length of an expression F = ∑_i=1^r G_i^k.

The answer to Problem <ref> is called the k-th Waring rank, or simply k-th rank, of F, and we denote it ^k_d(F). In this case, we need to consider the variety of k-th powers, i.e., 

    V_k,d = {[G^k] ∈ S_d  |  G ∈ S_d/k}.

That is, the variety obtained by considering the composition:

    π∘ν_k :  S_d/k→ S^k(S_d/k)  S_d,

where:

	
  * if W = S_d/k, then ν_k is the k-th Veronese embedding of W in S^kW;
	
  * if we consider the standard monomial basis w_α = ^α of W, i.e., |α| = d/k, then π is the linear projection from S^kW to S_d induced by the substitution w_α↦^α. In particular, we have that the center of the projection π is given by the homogeneous part of degree k of the ideal of the Veronese variety ν_d(^n).

Problem <ref> was considered by Fröberg, Shapiro and Ottaviani <cit.>. Their main result was that, if F is generic, then:

    _d^k(F) ≤ k^n,

i.e., the k^n-th secant variety of V_k,d fills the ambient space. This was proven by Terracini's lemma. Indeed, for any G,H ∈ S_d/k, we have that:

    . d/ dt|_t = 0(G+tH)^k = kG^k-1H;

therefore, we obtain that:

    T_[G^k] V_k,d = (⟨ [G^k-1H]  |  H ∈ S_d/k⟩),

and, by Terracini's lemma (Lemma <ref>), if Q is a generic point on ⟨ [G_1^k],…,[G_s^k]⟩, where the G_i's are generic forms of degree d/k, then: 

    T_Q σ_sV_k,d = ((G_1^k-1,…,G_s^k-1)_d).

In <cit.>, the authors showed that the family: 

    G_i_1,…,i_n = (x_0 + ξ^i_1x_1 + … + ξ^i_nx_n)^d/k∈ S_d/k,    for  i_1,…,i_n ∈{0,…,k-1},

where ξ^k = 1, is such that: 

    (G_1^k-1,…,G_s^k-1)_d = S_d.

In this way, they showed that σ_k^n(V_k,d) fills the ambient space. A remarkable fact with the upperbound (<ref>) is that it is independent of the degree of the polynomial, but it only depends on the power k. Now, the naive lower bound due to parameter counting is ⌈ S_d/ S_d/k⌉ = ⌈n+d  n/n+d/k  n⌉, which tends to k^n when d runs to infinity. 

In conclusion, we obtain that the main result of <cit.> is resumed as follows.

	Let F be a generic form of degree d in n+1 variables. Then,
	
    R^k_d(F) ≤ k^n.

	If d ≫ 0, then the latter bound is sharp.

This result gives an asymptotic answer to Problem <ref>, but, in general, it is not known for which degree d the generic k-th Waring rank starts to be equal to k^n, and it is not known what happens in lower degrees. 

We have explained in (<ref>) how to explicitly see the variety of powers V_k,d as a linear projection of a Veronese variety X_N,k = ν_k(^N), where N = n+d/k  n-1. It is possible to prove that σ_2(X_N,k) does not intersect the base of linear projection, and therefore, V_k,d is actually isomorphic to X_N,k. Unfortunately, higher secant varieties intersect non-trivially the base of the projection, and therefore, their images, i.e., the secant varieties of the varieties of powers, are more difficult to understand. However, computer experiments suggest that the dimensions are preserved by the linear projection; see <cit.> (Section 4) for more details about these computations (a Macaulay2 script with some examples is available in the ancillary files of the arXiv version of <cit.>). In other words, it seems that we can use the Alexander–Hirschowitz theorem to compute the dimensions of secant varieties of varieties of powers and provide an answer to Problem <ref>. More on this conjecture is explained in <cit.>.

	Let F be a generic form of degree d in n+1 variables. Then,
	
    ^k_d(F) = 
    		min{s ≥ 1  |  sn+d/2  n - s  2≥n+d  n}    for  k = 2; 
    min{s ≥ 1  |  sn+d/k  n≥n+d  n}    for  k ≥ 3.



	The latter conjecture claims that for k ≥ 3, the correct answer is given by the direct parameter count. For k = 2, we have that secant varieties are always defective. This is analogous to the fact that secant varieties to the two-fold Veronese embeddings are defective. Geometrically, this is motivated by Terracini's lemma and by the fact that:
	
    T_[G^2]V_2,d∩ T_[H^2] V_2,d = [GH],

	and not empty, as expected.



Here, we explain how the binary case can be treated; see <cit.>.
By (<ref>), the computation of the dimension of secant varieties of varieties of powers reduces to the computation of dimensions of homogeneous parts of particular ideals, i.e., their Hilbert functions. This relates Problem <ref> to some variation of Fröberg's conjecture, which claims that the ideal (G_1^k,…,G_s^k), where the G_i's are generic forms of degrees at least two, has Hilbert series equal to the right-hand side of (<ref>); see <cit.>. In the case of binary forms, by semicontinuity, we may specialize the G_i's to be powers of linear forms. In this way, we may employ the result of <cit.>, which claims that power ideals in two variables satisfy Fröberg–Iarrobino's conjecture, i.e., (<ref>) is actually an equality, and we conclude the proof of Conjecture <ref> in the case of binary forms.


By using an algebraic study on the Hilbert series of ideals generated by powers of forms, we have a complete answer to Problem <ref> in the following cases (see <cit.>):

	
  * binary forms (n = 1), where: 
	
    ^k_d(F) = ⌈d+1/d/k+1⌉;

	
  * ternary forms as sums of squares (n = 2, k = 2), where:
	
    ^2_d(F) = ⌈d+2  2/d/2+2  2⌉,

	except for d = 1,3,4, where ^2_d(F) = ⌈d+2  2/d/2+2  2⌉ +1.;
		
  * quaternary forms as sums of squares (n = 3, k = 2), where:
	
    ^2_d(F) = ⌈d+3  3/d/2+3  3⌉,

	except for d = 1,2, where ^2_d(F) = ⌈d+3  3/d/2+3  3⌉ +1.






§ BEYOND DIMENSIONS


We want to present here, as a natural final part of this work, a list of problems about secant varieties and decomposition of tensors, which are different from merely trying to determine the dimensions of the varieties σ_s(X) for the various X we have considered before. We will consider problems such as determining maximal possible ranks, finding bounds or exact values on ranks of given tensor, understanding the set of all possible minimal decompositions of a given tensor, finding equations for the secant varieties or studying what happens when working over ℝ. The reader should be aware of the fact that there are many very difficult open problems around these questions.



 §.§ Maximum Rank
 
A very difficult and still open problem is the one that in the Introduction we have called the “little Waring problem”. We recall it here.
 
Which is the minimum integer r such that any form can be written as a sum of r pure powers of linear forms?


This corresponds to finding the maximum rank of a form of certain degree d in a certain number n+1 of variables.

To our knowledge, the best general achievement on this problem is due to Landsberg and Teitler, who in <cit.> proved that the rank of a degree d form in n+1 variables is smaller than or equal to n+d  d-n. Unfortunately, this bound is sharp only for n=1 if d≥ 2 (binary forms); in fact, for example, if n=2 and d=3,4, then the maximum ranks are known to be 5<2+3 2-3=7 and 7<2+4 2-4=11, respectively; see <cit.>. Another general bound has been obtained by Jelisiejew <cit.>, who proved that, for F ∈ S^d^n+1, we have _ sym(F) ≤n+d-1  d-1 - n+d-5  d-3. Again, this bound is not sharp for n ≥ 2. Another remarkable result is the one due to Blekherman and Teitler, who proved in <cit.> that the maximum rank is always smaller than or equal to twice the generic rank. 

The latter inequality, which has a very short and elegant proof, holds also between maximal and generic X-ranks with respect to any projective variety X. 


In a few cases in small numbers of variables and small degrees, exact values of maximal ranks have been given. We resume them in the following table.



    d     n     Maximal Rank     Ref. 

binary forms     any     1     d     classical, <cit.> 

quadrics     2     any     n+1     classical 

plane cubics     3     2     5     <cit.> 

plane quartics     4     2     7     <cit.> 

plane quintics     5     2     10     <cit.> 

cubic surfaces     3     3     7     <cit.> 

cubic hypersurfaces     3     any     ≤⌊d^2 + 6d + 1/4⌋     <cit.>





We want to underline the fact that it is very difficult to find examples of forms having high rank, in the sense higher than the generic rank. Thanks to the complete result on monomials in <cit.> (see Theorem <ref>), we can easily see that in the case of binary and ternary forms, we can find monomials having rank higher than the generic one. However, for higher numbers of variables, monomials do not provide examples of forms of high rank. Some examples are given in <cit.>, and the spaces of forms of high rank are studied from a geometric point of view in <cit.>.



 §.§ Bounds on the Rank

In the previous subsection, we discussed the problem of finding the maximal rank of a given family of tensors. However, for a given specific tensor T, it is more interesting, and relevant, to find explicit bounds on the rank of T itself. For example, by finding good lower and upper bounds on the rank of T, one can try to compute actually the rank of T itself, but usually, the maximal rank is going to be too large to be useful in this direction.

One typical approach to find upper bounds is very explicit: by finding a decomposition of T. In the case of symmetric tensors, that is in the case of homogeneous polynomials, the apolarity lemma (Lemma <ref>) is an effective tool to approach algebraically the study of upper bounds: by finding the ideal of a reduced set of points 𝕏 inside F^⊥, we bound the rank of F from above by the cardinality of 𝕏.


For F=x_0x_1x_2, in standard notation, we have F^⊥=(y_0^2,y_1^2,y_3^2), and we can consider the complete intersection set of four reduced points 𝕏 whose defining ideal is (y_1^2-y_0^2, y_2^2-y_0^2), and thus, the rank of F is at most four. Analogously, if F = x_0x_1^2x_2^3, we have F^⊥ = (y_0^2,y_1^3,y_3^4), and we can consider the complete intersection of 12 points defined by (y_1^3-y_0^3,y_2^4-y_0^4). 


Other upper bounds have been given by using different notions of rank.

We say that a scheme Z⊂ℙ^N is curvilinear if it is a finite union of schemes of the form 𝒪_𝒞_i,P_i/𝔪_P_i^e_i, for smooth points P_i on reduced curves 𝒞_i⊂ℙ^N. Equivalently, the tangent space at each connected component of Z supported at the P_i's has Zariski dimension ≤ 1. The curvilinear rank _ curv(F) of a degree d form F in n+1 variables is:

    _ curv(F):=min{(Z)  |   Z⊂ X_n,d,   Z  curvilinear,  [F]∈⟨ Z ⟩}.


 With this definition, in <cit.>, it is proven that the rank of an F∈ S^d^n+1 is bounded by (_ curv(F)-1)d+2-_ curv(F). This result is sharp if _ curv(F) =2, 3; see <cit.>.

Another very related notion of rank is the following; see <cit.>.

We define the smoothable rank of a form F ∈ S^d^n+1 as:

    _ smooth(F) := min{ (Z)  |  Z  is a limit of smooth schemes  Z_i  such that
     
    Z,Z_i⊂ X_n,d,  are zero-dim schemes with (Z_i)=(Z), 
     and  [F]∈⟨ Z⟩}.


In <cit.> (Section 2), it is proven that if F is a ternary form of degree d, then _ sym(F) ≤ (_ smooth(F)-1)d. We refer to <cit.> for a complete analysis on the relations between different notions of ranks. 

The use of the apolarity lemma (Lemma <ref>) to obtain lower bounds to the symmetric-rank of a homogeneous polynomial was first given in <cit.>.


If the ideal F^⊥ is generated in degree t and 𝕏 is a finite scheme apolar to F, that is I_𝕏⊂ F^⊥, then:

    1/t F^⊥≤𝕏.


This result is enough to compute the rank of the product of variables.

For F=x_0x_1x_2, Theorem <ref> yields:

    1  2 8≤𝕏.

If we assume  to be reduced, i.e., = ||, by the apolarity lemma, we get _ sym(F) ≥ 4, and thus, by Example <ref>, the rank of F is equal to four. However, for the monomial x_0x_1^2x_2^3, we get the lower bound of six, which does not allow us to conclude the computation of the rank, since Example <ref> gives us 12 as the upper bound.

To solve the latter case, we need a more effective use of the apolarity lemma in order to produce a better lower bound for the rank; see <cit.>.


Let F be a degree d form, and let e > 0 be an integer. Let I be any ideal generated in degree e, and let G be a general form in I. For s ≫ 0, we have:

    _ sym(F) ≥1/e∑_i=0^s (R/(F^⊥ : I+(G)),i).


A form for which there exists a positive integer e such that the latter lower bound is actually sharp is called e-computable; see <cit.>. Theorem <ref> was first presented in <cit.> in the special case of e = 1: this was the key point to prove Theorem <ref> on the rank of monomials, by showing that monomials are one-computable. In order to give an idea of the method, we give two examples: in the first one, we compute the rank of x_0x_1^2x_2^3 by using one-computability, while in the second one, we give an example in which it is necessary to use two-computability; see <cit.>.


Consider again F=x_0x_1^2x_2^3. We use Theorem <ref> with e = 1, G=y_0 and I=(y_0). Note that:

    F^⊥ : I + (G) = (y_0^2,y_1^3,y_2^4) : (y_0) + (y_0) = (y_0,y_1^3,y_2^4).

This yields to: 

    _ sym(F) ≥∑_i=0^s (R/(y_0,y_1^3,y_2^4),i)=12,

since (R/(y_0,y_1^3,y_2^4),z) = 1 + 2z + 3z^2 + 3z^3 + 2z^4 + z^5. Hence, by using Example <ref>, we conclude that the rank of F is actually 12.

 

Consider the polynomial:

    [ F = x_0^11 -22x_0^9x_1^2 + 33x_0^7x_1^4 -22x_0^9x_2^2 + 396x_0^7x_1^2x_2^2-462x_0^5x_1^4x_2^2 +;                                          33x_0^7x_2^4 -462x_0^5x_1^2x_2^4 + 385x_0^3x_1^4x_2^4, ],

we show that F is two-computable and _ sym(F)=25.
By direct computation, we get: 
    F^⊥ = ((y_0^2 +y_1^2+y_2^2)^2, G_1, G_2),

where G_1=y_1^5 +y_2(y_0^2 +y_1^2+y_2^2)^2 and G_2=y_2^5 + y_0(y_0^2 +y_1^2+y_2^2)^2.

Hence, by (<ref>), we get:

    _ sym(F)≥1/2∑_i=0^∞ HF(T/ (F^⊥ : (y_0^2 +y_1^2+y_2^2) + (y_0^2 + y_1^2+y_2^2)),i)= 25.

Moreover, the ideal (G_1,G_2) ⊂ F^⊥ is the ideal of 25 distinct points, and thus, the conclusion follows. It can be shown that F is not one-computable; see <cit.> (Example 4.23).


Another way to find bounds on the rank of a form is by using the rank of its derivatives. A first easy bound on the symmetric-rank of a homogeneous polynomial F∈[x_0,…, x_n] (where  is a characteristic zero field) is directly given by the maximum between the symmetric-ranks of its derivatives; indeed, if F = ∑_i=1^r L_i^d, then, for any j = 0,…,n,

    ∂ F/∂ x_j = ∑_i=1^r ∂ L_i^d/∂ x_j = (d-1)∑_i=1^r ∂ L_i/∂ x_j L_i^d-1.

A more interesting bound is given in <cit.>.

Let 1≤ p≤ n be an integer, and let F∈[x_0,…, x_n] be a form, where  is a characteristic zero field. Set F_k=∂ F/∂ x_k, for 0≤ k≤ n. If:

    rk_ sym(F_0 +∑_k=1^nλ_k F_k)≥ m,

for all λ_k∈, and if the forms F_1, F_2, … ,F_p are linearly independent, then:

    rk_ sym(F) ≥ m + p.


The latter bound was lightly improved in <cit.>.





Formula (<ref>) can be generalized to higher order differentials. As a consequence, for any G ∈ S^j V^*, with 1 ≤ j ≤ d-1, we have that _ sym(F) ≥_ sym(G∘ F), and in particular, if F ∈⟨ L_1^d,…,L_r^d⟩, we have that
G∘ F ∈⟨ L_1^d-j,…,L_r^d-j⟩.
Since this holds for any G ∈ S^jV^*, we conclude that the image of the (j,d-j)-th catalecticant matrix is contained in ⟨ L_1^d-j,…,L_r^d-j⟩. Therefore, 

    _ sym(F) ≥_( Imm Cat_j,d-j(F)) = rk Cat_j,d-j(F).

The latter bound is very classical and goes back to Sylvester. By using the geometry of the hypersurface V(F) in ^n, it can be improved; see <cit.>.

Let F be a degree d form with n+1 essential variables. Let 1 ≤ j ≤ d-1. Use the convention that ∅ = -1. Then, the symmetric-rank of F is such that:

    _ sym(F) ≥rk Cat_j,d-j(F) + Σ_j(F) + 1,

where Cat_j,d-j(F) is the (j,d-j)-th catalecticant matrix of F and:

    Σ_j(F)={ P∈ V(F) ⊂ V : ∂^α F∂^α(P)=0,   ∀ |α| ≤ j }.


The latter result has been used to find lower bounds on the rank of the determinant and the permanent of the generic square matrix; see <cit.> (Corollary 1.4).

The bound (<ref>) given by the ranks of catalecticant matrices is a particular case of a more general fact, which holds for general tensors. 

Given a tensor T ∈ V_1⊗…⊗ V_d, there are several ways to view it as a linear map. For example, we can “reshape” it as a linear map V^*_i → V_1⊗…⊗V_i⊗…⊗ V_d, for any i, or as V^*_i⊗ V^*_j → V_1⊗…⊗V_i⊗…⊗V_j⊗…⊗ V_d, for any i ≠ j, or more in general, as: 

    V_i_1^*⊗ V^*_i_2⊗…⊗ V_i_s^* → V_1⊗…⊗V_i_1⊗…⊗V_i_s⊗…⊗ V_d,

for any choice of i_1,…,i_s. All these ways of reshaping the tensor are called flattenings. Now, if T is a tensor of rank r, then all its flattenings have (as matrices) rank at most r. In this way, the ranks of the flattenings give lower bounds for the rank of a tensor, similarly as the ranks of catalecticant matrices gave lower bounds for the symmetric-rank of a homogeneous polynomial. 

We also point out that other notions of flattening, i.e., other ways to construct linear maps starting from a given tensor, have been introduced in the literature, such as Young flattenings (see <cit.>) and Koszul flattenings (see <cit.>). These were used to find equations of certain secant varieties of Veronese and other varieties and to provide algebraic algorithms to compute decompositions.

We conclude this section with a very powerful method to compute lower bounds on ranks of tensors: the so-called substitution method. In order to ease the notation, we report the result in the case T ∈ V_1⊗ V_2 ⊗ V_3, with _ V_i = n_i. For a general result, see <cit.> (Appendix B).


	Let T ∈ V_1⊗ V_2 ⊗ V_3. Write T = ∑_i=1^n_1 e_i ⊗ T_i, where the e_i's form a basis of V_1 and the T_i's are the corresponding “slices” of the tensor. Assume that M_n_1≠ 0. Then, there exist constants λ_1,…,λ_n_1 - 1 such that the tensor:
	
    T' = ∑_i=1^n_1-1 e_i ⊗ (T_i - λ_i T_n_1) ∈^n_1-1⊗ V_2 ⊗ V_3,

	has rank at most (T) - 1. If T_n_1 is a matrix of rank one, then equality holds.

Roughly speaking, this method is applied in an iterative way, with each of the V_i's playing the role of V_1 in the theorem, in order to reduce the tensor to a smaller one whose rank we are able to compute. Since, in the theorem above, (T) ≥(T') + 1, at each step, we get a plus one on the lower bound. For a complete description of this method and its uses, we refer to <cit.> (Section 5.3).

A remarkable use of this method is due to Shitov, who recently gave counterexamples to very interesting conjectures such as Comon's conjecture, on the equality between the rank and symmetric-rank of a symmetric tensor, and Strassen's conjecture, on the additivity of the tensor rank for sums of tensors defined over disjoint subvector spaces of the tensor space.

We will come back with more details on Strassen's conjecture, and its symmetric version, in the next section. We spend a few words more here on Comon's conjecture.

Given a symmetric tensor F ∈ S^dV ⊂ V^⊗ d, we may regard it as a tensor, forgetting the symmetries, and we could ask for its tensor rank, or we can take into account its symmetries and consider its symmetric-rank. Clearly, 

    (F) ≤_ sym(F).

The question raised by Comon asks if whether such an inequality is actually an equality. Affirmative answers were given in several cases (see <cit.>). In <cit.>, Shitov found an example (a cubic in 800 variables) where the inequality (<ref>) is strict. As the author says, unfortunately, no symmetric analogs of this substitution method are known. However, a possible formulation of such analogs, which might lead to a smaller case where (<ref>) is strict, was proposed.

	Let F,G ∈ S = [x_0,…,x_n] of degree d,d-1, respectively. Let L be a linear form. Then,
	
    _ sym(F+LG) ≥ d + min_L' ∈ S_1_ sym(F+L'G).



	A symmetric tensor F ∈ S^dV can be viewed as a partially-symmetric tensor in S^d_1V⊗…⊗ S^d_mV, for any d = (d_1,…,d_m) ∈^m such that d_1+…+d_m = d. Moreover, if d' = (d'_1,…,d'_m') ∈^m' is a refinement of d, i.e., there is some grouping of the entries of d' to get d, then we have:
	
    _ sym(F) ≥_d(F) ≥_d'(F),

	which is a particular case of (<ref>). In the recent paper <cit.>, the authors investigated the partially-symmetric version of Comon's question, i.e., the question if, for a given F, (<ref>) is an equality or not. Their approach consisted of bounding from below the right-hand side of (<ref>) with the simultaneous rank of its partial derivatives of some given order and then studying the latter by using classical apolarity theory (see also Remark <ref>). If such a simultaneous rank coincides with the symmetric-rank of F, then also all intermediate ranks are the same. In particular, for each case in which Comon's conjecture is proven to be true, then also all partially-symmetric tensors coincide. For more details, we refer to <cit.>, where particular families of homogeneous polynomials are considered. 




 §.§ Formulae for Symmetric Ranks

In order to find exact values of the symmetric-rank of a given polynomial, we can use one of the available algorithms for rank computations; see Section <ref>. However, as we already mentioned, the algorithms will give an answer only if some special conditions are satisfied, and the answer will be only valid for that specific form. Thus, having exact formulae working for a family of forms is of the utmost interest.

Formulae for the rank are usually obtained by finding an explicit (a posteriori sharp) upper bound and then by showing that the rank cannot be less than the previously-found lower bound.

An interesting case is the one of monomials. The lower bound of (<ref>) is used to obtain a rank formula for the complex rank of any monomial, similarly as in Example <ref>; i.e., given a monomial F = ^α, whose exponents are increasingly ordered, we have that F^⊥ = (y_0^α_0+1,…,y_n^α_n+1), and then, one has to:


  * first, as in Example <ref>, exhibit the set of points apolar to F given by the complete intersection (y_1^α_1-y_0^α_1,…,y_n^α_n - y_0^α_0); this proves that the right-hand side of (<ref>) is an upper bound for the rank;

  * second, as in Example <ref>, use Theorem <ref> with e = 1 and G = y_0 to show that the right-hand side of (<ref>) is a lower bound for the rank.


Let 1 ≤α_0 ≤α_1 …≤α_n. Then,

    _ sym(^α) = 1/α_0+1∏_i=0^n (α_i+1).



Another relevant type of forms for which we know the rank is the one of reduced cubic forms. The reducible cubics, which are not equivalent to a monomial (up to change of variables), can be classified into three canonical forms. The symmetric complex rank for each one was computed, as the following result summarizes: the first two were first presented in <cit.>, while the last one is in <cit.>. In particular, for all three cases, we have that the lower bound given by Theorem <ref> is sharp.

Let F∈ℂ[x_0,…,x_n] be a form essentially involving n +1 variables, which is not equivalent to a monomial. If F is a reducible cubic form, then one and only one of the following holds:


  * F is equivalent to:

    x_0(x_0^2+ x_1^2+… + x_n^2),

and _ sym(F)=2n.

  * F is equivalent to:

    x_0(x_1^2+ x_2^2+ … + x_n^2),

and _ sym(F)=2n.

  * F equivalent to:

    x_0(x_0x_1 + x_2x_3 + x_4^2+ … + x_n^2),

and _ sym(F)=2n +1.







Another way to find formulae for symmetric-ranks relies on a symmetric version of Strassen's conjecture on tensors. In 1973, Strassen formulated a conjecture about the additivity of the tensor ranks <cit.>, i.e., given tensors T_i,…,T_s in V^⊗ d defined over disjoint subvector spaces, then,

    (∑_i=1^s T_i)=∑_i = 1^s (T_i).

After a series of positive results (see, e.g., <cit.>), Shitov gave a proof of the existence of a counter-example to the general conjecture in the case of tensors of order three <cit.>. Via a clever use of the substitution method we introduced in the previous section, the author described a way to construct a counter-example, but he did not give an explicit one.

However, as the author mentioned is his final remarks, no counter example is known for the symmetric version of the conjecture that goes as follows: given homogeneous polynomials F_1,…,F_s in different sets of variables, then:

    _ sym(∑_i = 1^s F_i) = ∑_i = 1^s _ sym(F_i).
 
In this case, Strassen's conjecture is known to be true in a variety of situations. The case of sums of coprime monomials was proven in <cit.> via apolarity theory by studying the Hilbert function of the apolar ideal of F = ∑_i=1^s F_i. Indeed, it is not difficult to prove that: 

    F^⊥ = ⋂_i=1^s F_i^⊥ + (F_i + λ_i,j F_j     :  i ≠ j), 
        where the λ_i,j's are suitable coefficients.

In this way, since apolar ideals of monomials are easy to compute, it is possible to express explicitly also the apolar ideal of a sum of coprime monomials. Therefore, the authors applied an analogous strategy as the one used for Theorem <ref> (by using more technical algebraic computations) to prove that Strassen's conjecture holds for sums of monomials. 

In <cit.>, the authors proved that Strassen's conjecture holds whenever the summands are in either one or two variables. In <cit.>, the authors provided conditions on the summands to guarantee that additivity of the symmetric-ranks holds. For example, in <cit.>, the author showed that whenever the catalecticant bound (<ref>) (or the lower bound given by Theorem <ref>) is sharp for all the F_i's, then Strassen's conjecture holds, and the corresponding bound for ∑_i=1^s F_i is also sharp. 

A nice list of cases in which Strassen's conjecture holds was presented in <cit.>. This was done again by studying the Hilbert function of the apolar ideal of F = ∑_i=1^s F_i, computed as described in (<ref>), and employing the bound given by Theorem <ref>.


Let F = F_1 +…+ F_m, where the degree d forms F_i are in different sets of variables. If, for i = 1,…,m, each F_i is of one of the following types:


  * F_i is a monomial;

  * F_i is a form in one or two variables;

  * Fi =x_0^a(x_1^b+… +x_n^b) with a+1≥ b;

  * Fi =x_0^a(x_1^b+x_2^b);

  * Fi =x_0^a(x_0^b+… +x_n^b) with a+1≥ b;

  * Fi =x_0^a(x_0^b+x_1^b+x_2^b);

  * F_i=x_0^aG(x_1,…,x_n) where G^⊥=(H_1,…,H_m) is a complete intersection and a≤ H_i;

  * F_i is a Vandermonde determinant;

then Strassen's conjecture holds for F.






 §.§ Identifiability of Tensors

For simplicity, in this section we work on the field ℂ of the complex numbers. Let us consider tensors in = ℂ^n_1+1⊗…⊗ℂ^n_d+1. A problem of particular interest when studying minimal decompositions of tensors is to count how many there are.

 
Suppose a given tensor T ∈ has rank r, i.e., it can be written as T = ⊕_i=1^r v_i^1⊗…⊗v_i^d. When is it that such a decomposition is unique (up to permutation of the summands and scaling of the vectors)? 


This problem has been studied quite a bit in the last two centuries (e.g., see <cit.>), and it is also of interest with respect to many applied problems (e.g., see <cit.>). Our main references for this brief exposition are <cit.>.

Let us begin with a few definitions.

 A rank-r tensor T ∈ is said to be identifiable over ℂ if its presentation 
T = ⊕_i=1^r v_i^1⊗…⊗v_i^d
is unique (up to permutations of the summands and scaling of the vectors).

It is interesting to study the identifiability of a generic tensor of given shape and rank.

We say that tensors in  are r-generically identifiable over ℂ if identifiability over ℂ holds in a Zariski dense open subset of the space of tensors of rank r. Moreover, we say that the tensors in  are generically identifiable if they are r_g-generically identifiable, where r_g denotes the generic rank in .

Let us recall that the generic rank for tensors  is the minimum value for which there is a Zariski open non-empty set U of  where each point represents a tensor with rank ≤ r_g; see Section <ref>.




Considering 𝐧= (n_1,… , n_d), let X_n⊂ℙ be the Segre embedding of ^n_1×…×^n_d. As we already said previously, if r_g is the generic rank for tensors in , then σ_r_g(X_n) is the first secant variety of X_, which fills the ambient space. Therefore, to say that the tensors in  are generically identifiable over ℂ amounts to saying the following: let r_g be the generic rank with respect to the Segre embedding X_ in ; then, for the generic point [T] ∈ℙ, there exists a unique ℙ^r-1, which is r_g-secant to X_n in r_g distinct points and passes through [T]. The r_g points of X_n gives (up to scalar) the r_g summands in the unique (up to permutation of summands) minimal decomposition of the tensor T.

When σ_r(X_n)≠ℙ, i.e., the rank r is smaller than the generic one (we can say that r is sub-generic), then we have that the set of tensors T ∈ with rank r is r
-generically identifiable over ℂ if there is an open set U of σ_r(X_n) such that for the points [T] in U, there exists a unique ℙ^r-1, which is r-secant in r distinct points to X_n and passes through [T].


Obviously, the same problem is interesting also when treating symmetric or skew-symmetric tensors, i.e., when n_1=… = n_d=n and T ∈ S^d(ℂ^n+1) ⊂ or T ∈⋀^d(ℂ^n+1). From a geometric point of view, in these cases, we have to look at Veronese varieties or Grassmannians, respectively, and their secant varieties, as we have seen in the previous sections.


Generic identifiability is quite rare as a phenomenon, and it has been largely investigated; in particular, we refer to <cit.>. As an example of how generic identifiability seldom presents itself, we can consider the case of symmetric tensors.

It is classically known that there are three cases of generic identifiability, namely:


  * binary forms of odd degree (n = 1 and d = 2t+1), where the generic rank is t+1 <cit.>;

  * ternary quintics (n = 2 and d = 5), where the generic rank is seven <cit.>;

  * quaternary cubics (n = 3 and d = 3), where the generic rank is five <cit.>.

Recently, Galuppi and Mella proved that these are the only generically identifiable cases when considering symmetric-ranks of symmetric tensors; see <cit.>.


When we come to partially-symmetric tensors (which are related to Segre–Veronese varieties, as we described in Section <ref>), a complete classification of generically-identifiable cases is not known, but it is known that it happens in the following cases; see <cit.>.


  * S^d_1^2 ⊗…⊗ S^d_t^2, with d_1 ≤… d_t and d_1+1 ≥ r_g, where r_g is the generic partially-symmetric-rank, i.e., forms of multidegree (d_1,…,d_t) in t sets of two variables; here, the generic partially-symmetric-rank is t+1 <cit.>;

  * S^2^n+1⊗ S^2^n+1, i.e., forms of multidegree (2,2) in two sets of n+1 variables; her,e the generic partially-symmetric-rank is n+1 <cit.>;

  * S^2^3 ⊗ S^2^3 ⊗ S^2^3 ⊗ S^2^3, i.e., forms of multidegree (2,2,2,2) in four sets of three variables; here, the generic partially-symmetric-rank is four (this is a classical result; see also <cit.>);

  * S^2^3 ⊗ S^3^3, i.e., forms of multidegree (2,3) in two sets of three variables; here, the generic partially-symmetric-rank is four <cit.>;

  * S^2^3 ⊗ S^2^3 ⊗ S^4^3, i.e., forms of multidegree (2,2,4) in three sets of three variables; here, the generic partially-symmetric-rank is seven <cit.>.


When considering r-generically identifiable tensors for sub-generic rank, i.e., for r < r_g, things change completely, in as much as we do expect r-generically-identifiability in this case. Again, the symmetric case is the best known; in <cit.>, it was proven that every case where r is a sub-generic rank and σ_r(X_n,d) has the expected dimension for the Veronese variety X_n,d, r-generically identifiability holds with the only following exceptions:


  * S^6^3, i.e., forms of degree six in three variables, having rank nine;

  * S^4^4, i.e., forms of degree four in four variables, having rank eight;

  * S^3^6, i.e., forms of degree three in six variables, having rank nine.

In all the latter cases, the generic forms have exactly two decompositions.


Regarding generic identifiability for skew-symmetric tensors, there are not many studies, and we refer to <cit.>.


It is quite different when we are in the defective cases, namely, when we want to study r-generic identifiability and the r-th secant variety is defective. In this case, non-identifiability is expected; in particular, we will have that the number of decompositions for the generic tensor parametrized by a point of σ_r(Y) is infinite.



 §.§ Varieties of Sums of Powers

Identifiability deals with the case in which tensors have a unique (up to permutation of the summands) decomposition. When the decomposition is not unique, what can we say about all possible decompositions of the given tensor? In the case of symmetric tensors, that is homogeneous polynomial, an answer is given by studying varieties of sums of powers, the so-called , defined by Ranestad and Schreyer in <cit.>.

 
Let F be a form in n+1 variables having Waring rank r, and let Hilb_rℙ^n be the Hilbert scheme of r points in ℙ^n; we define:

    VSP(F,r)={𝕏 = {P_1,…,P_r}∈Hilb_rℙ^n:I_𝕏 = ℘_1∩…∩℘_r⊂ F^⊥}.



For example, when identifiability holds,  is just one single point. It is interesting to note that, even for forms having generic rank, the corresponding  might be quite big, as in the case of binary forms of even degree.

Using Sylvester's algorithm we have a complete description of  for binary forms, and it turns out to be always a linear space.

 
Consider the binary form F=x_0^2x_1^2. Since F^⊥=(y_0^3,y_1^3), by Sylvester's algorithm, we have that the rank of F is three. Moreover, by the apolarity lemma, we have that (F,3) is the projectivization of the vector space W = ⟨ y_0^3,y_1^3⟩ because the generic form in W has three distinct roots.


In general, the study of VSPs
 is quite difficult, but rewarding: VSPs play an important role in classification work by Mukai see <cit.>. For a review of the case of general plane curves of degree up to ten, that is for general ternary forms of degree up to ten, a complete description is given in <cit.>, including results from Mukai and original results. We summarize them in the following. 


Let F∈ S^d^3 be a general ternary cubic with d = 2t - 2, 2 ≤ t ≤ 5, then:

    (F, t+1 2)≃ G(t, V, η) = {E ∈ G(t,V)  | ∧^2E ⊂η}

where V is a 2t + 1-dimensional vector space and η is a net of alternating forms η: Λ^2 V →ℂ^3 on V. Moreover:


  * if F is a smooth plane conic section, then (F,3) is a Fano three-fold of index two and degree five in ℙ^6.

  * if F is a general plane quartic curve, then (F, 6) is a smooth Fano three-fold of index one and genus 12 with anti-canonical embedding of degree 22;

  * if F is a general plane sextic curve, then (F, 10) is isomorphic to the polarized K3-surface of genus 20;

  * if F is a general plane octic curve, then (F,15) is finite of degree 16, i.e., consists of 16 points.



Very often, for a given specific form, we do not have such a complete description, but at least, we can get some relevant information, for example about the dimension of the VSP: this is the case for monomials.


Let F∈[x_0, … , x_n] be a monomial F = ^α with exponents 0 < α_0 ≤…≤α_n. Let A = [y_0,…, y_n]/(y_1^α_1+1,…,y_n^α_n+1). Then, (F,_ sym(F)) is irreducible and: 
    (F,_ sym(F)) = ∑_i=1^n(A;d_i - d_0).




A complete knowledge of (F,r) gives us a complete control on all sums of powers decompositions of F involving r summands. Such a complete knowledge comes at a price: a complete description of the variety of sums of powers might be very difficult to obtain. However, even less complete information might be useful to have and, possibly, easier to obtain. One viable option is given by Waring loci as defined in <cit.>.


The Waring locus of a degree d form F ∈ S^dV is:

    𝒲_F={ [L]∈ℙV  :  F=L^d+L_2^d+…+L_r^d,  r = _ sym(F)},

i.e., the space of linear form which appears in some minimal sums of powers decomposition of F. The forbidden locus of F is defined as the complement of the Waring locus of F, and we denote it by ℱ_F.



	In this definition, the notion of essential variables has a very important role; see Remark <ref>. In particular, it is possible to prove that if F ∈[x_0,…,x_n] has less than n+1 essential variables, say x_0,…,x_m, then for any minimal decomposition F = ∑_i=1^r L_i, the L_i's also involve only the variables x_0,…,x_m. For this reason, if in general, we have F ∈ S^dV, which has less than _ V essential variables, say that W ⊂ V is the linear span of a set of essential variables, then 𝒲_F ⊂ W. 


In <cit.>, the forbidden locus, and thus the Waring locus, of several classes of polynomials was computed. For example, in the case of monomials, we have the following description.

 
If F = ^α, such that the exponents are increasingly ordered and m = min_i{α_i = α_0}, then:

    ℱ_F = V(y_0⋯ y_m).



The study of Waring and forbidden loci can have a two-fold application. One is to construct step-by-step a minimal decomposition of a given form F: if [L] belongs to 𝒲_F, then there exists some coefficient λ∈ such that F' = F+λ L^d has rank smaller than F; then, we can iterate the process by considering 𝒲_F'. In <cit.>, this idea is used to present an algorithm to find minimal decompositions of forms in any number of variables and of any degree of rank r ≤ 5 (the analysis runs over all possible configuration of r points in the space whose number of possibilities grows quickly with the rank); a Macaulay2 package implementing this algorithm can be found in the ancillary files of the arXiv version of <cit.>. A second possible application relates to the search for forms of high rank: if [L] belongs to 𝒲_F, then the rank of F + λ L^d cannot increase as λ varies, but conversely, if the rank of F+λ L^d increases, then [L] belongs to the forbidden locus of F. Unfortunately, it is not always possible to use elements in the forbidden locus to increase the rank of a given form; however, this idea can give a place to look for forms of high rank. For example, since ℱ_x_0x_1x_2=V(y_0y_1y_2), that is the forbidden locus of the monomial F=x_0x_1x_2 is the union of the three coordinate lines of ^2, the only possible way to make the rank of F increase is to consider F+λ L^3 where L is a linear form not containing at least one of the variables. However, as some computations can show, the rank of F+λ L^3 does not increase for any value of λ and for any choice of L in the forbidden locus. 


Another family of polynomial for which we have a complete description of Waring and forbidden loci are binary forms.


Let F be a degree d binary form, and let G ∈ F^⊥ be an element of minimal degree. Then,


  * if rk(F) <
d+1 2, then 𝒲_F=V(G);

  * if rk(F) >
d+1  2, then ℱ_F=V(G);

  * if rk(F) =
d+1  2 and d is even, then ℱ_F is finite and not empty; 

if rk(F) =
d+1  2 and d is odd, then 𝒲_F=V(G).




The result about Waring and forbidden loci of binary forms can be nicely interpreted in terms of rational normal curves. If F=x_0x_1^2, then:

    ℱ_F={ [y_1^3] },

and this means that any plane containing [F] and [y_1^3] is not intersecting the twisted cubic curve in three distinct points; indeed, the line spanned by [F] and [y_1^3] is tangent to the twisted cubic curve.

Other families of homogeneous polynomials for which we have a description of Waring and forbidden loci are quadrics <cit.> (Corollary 3.2) (in which case, the forbidden locus is given by a quadric) and plane cubics <cit.> (Section 3.4). We conclude with two remarks coming from the treatment of the latter case:


  * in all the previous cases, the Waring locus of F is always either closed or open. However, this is not true in general. In fact, for the cusp F=x_0^3+x_1^2x_2, we have that the Waring locus is: 
 
    𝒲_F={[y_0^3]}∪{[(ay_1+by_2)^3] : a,b∈ℂ b≠ 0},

 that is the Waring locus is given by two disjoint components: a point and a line minus a point; therefore, 𝒲_F is neither open nor closed in ℙ^2;

  * since the space of minimal decompositions of forms of high rank is high dimensional, it is expected that the Waring locus is very large, and conversely, the forbidden locus is reasonably small. For example, the forbidden locus of the maximal rank cubic F=x_0(x_1^2+x_0x_2) is just a point, i.e., 
  
    ℱ_F={[y_0^3]}.

There is an open conjecture stating that, for any form F, the forbidden locus ℱ_F is not empty.






 §.§ Equations for the Secant Varieties

A very crucial problem is to find equations for the secant varieties we have studied in the previous sections, mainly for Veronese, Segre and Grassmann varieties. Notice that having such equations (even equations defining only set-theoretically the secant varieties in question) would be crucial in having methods to find border ranks of tensors.



  §.§.§ Segre Varieties

Let us consider first Segre varieties; see also <cit.>. In the case of two factors, i.e., X_𝐧 with 𝐧 =(n_1,n_2), the Segre variety, which is the image of the embedding:

    ℙ^n_1×ℙ^n_2→ X_𝐧⊂ℙ^N ,   N = (n_1 + 1)(n_2 + 1)-1,

corresponds to the variety of rank one matrices, and σ_s(X_𝐧) corresponds to the variety of rank s matrices, which is defined by the ideal generated by the (s + 1)× (s + 1) minors of the generic (n_1 + 1)× (n_2 + 1) matrix, whose entries are the homogenous coordinates of ℙ^N. In this case, the ideal is rather well understood; see, e.g., <cit.> and also the extensive bibliography given in the book of Weyman <cit.>.

We will only refer to a small part of this vast subject, and we recall that the ideal I_σ_s(X_𝐧) is a perfect ideal of height (n_1+1-(s + 1)-1)× (n_2+1-(s + 1)-1) = (n_1-s-1) × (n_2-s-1) in the polynomial ring with N + 1 variables, with a very well-known resolution: the Eagon–Northcott complex. It follows from this description that all the secant varieties of the Segre embeddings of a product of two projective spaces are arithmetically Cohen–Macaulay varieties. Moreover, from the resolution, one can also deduce the degree, as well as other significant geometric invariants, of these varieties. A determinantal formula for the degree was first given by Giambelli. There is, however, a reformulation of this result, which we will use (see, e.g., <cit.> (p. 244) or <cit.>), where this lovely reformulation of Giambelli's Formula is attributed to Herzog and Trung:

    (σ_s(X_(n_1,n_2))) = ∏_i=0^n_1-sn_2+1+i s/s+i s
    .

Let us now pass to the case of the Segre varieties with more than two factors. Therefore, let X_𝐧⊂ℙ^N with 𝐧 = (n_1,… ,n_t), N = Π_i=1^t(n_i + 1)-1 and t ≥ 3, where we usually assume that n_1 ≥…≥ n_t.

If we let T be the generic (n_1 +1)×…×(n_t +1) tensor whose entries are the homogeneous coordinates in ℙ^N, then it is well known that the ideal of X_𝐧 has still a determinantal representation, namely it is generated by all the 2 × 2 “minors” of T, that is the 2× 2 minors of the flattenings of T. It is natural to ask if the flattenings can be used also to find equations of higher secant varieties of X_𝐧. If we split 1,…, t into two subsets, for simplicity say 1,…, ℓ and ℓ+1,…, t, then we can form the composition:

    ν_(n_1,…,n_ℓ)×ν_(n_ℓ+1,…,n_t) : (ℙ^n_1×…×ℙ^n_ℓ ) × (ℙ^n_ℓ +1×…×ℙ^n_t) →ℙ^a ×ℙ^b,

where a = Π_i=1^ℓ(n_i + 1)-1, b=Π_i=ℓ+1^t(n_i + 1)-1, followed by:

    ν_1,1: ℙ^a ×ℙ^b →ℙ^N,      N   as above.

Clearly X_𝐧⊂ν_1,1(ℙ^a ×ℙ^b), and hence,
σ_s(X_𝐧) ⊂σ_s(ν_1,1(ℙ^a ×ℙ^b)).

Thus, the (s + 1)× (s + 1) minors of the matrix associated with the embedding ν_1,1 will all vanish on σ_s(X_𝐧). That matrix, written in terms of the coordinates of the various ℙ^n_i, is what we have called a flattening of the tensor T.

As we have seen in (<ref>), we can perform a flattening of T for every partition of 1,…,t into two subsets. The (s + 1) × (s + 1) minors of all of these flattenings will give us equations that vanish on σ_s(X_𝐧). In <cit.>, it was conjectured that,
at least for s = 2, these equations are precisely the generators for the ideal I_σ_2(X_𝐧) of σ_2(X_𝐧). The conjecture was
proven in <cit.> for the special case of t = 3 (and set theoretically for all t's). Then, Allman and Rhodes <cit.> proved the conjecture for up to five factors, while Landsberg and Weyman <cit.> found the generators for the defining ideals of secant varieties for the Segre varieties in the following cases: all secant
varieties for ℙ^1 ×ℙ^m ×ℙ^n for all m, n; the secant line varieties of the Segre varieties with four factors; the secant
plane varieties for any Segre variety with three factors. The proofs use representation theoretic methods. 


Note that for s > 2, one cannot expect, in general, that the ideals I_σ_s(X_𝐧) are generated by the (s+1)×(s+1) minors of flattenings of T. Indeed, in many cases, there are no such minors, e.g., it is easy to check that if we consider 𝐧=(1,1,1,1,1) and X_𝐧⊂ℙ^31, we get that the flattenings can give only ten 4× 8 matrices and five 2× 16 matrices. Therefore, we get quadrics, which generate I_X_𝐧, and quartic forms, which are zero on σ_3(X_𝐧), but no equations for σ_4(X_𝐧) or σ_5(X_𝐧), which, by a simple dimension count, do not fill all of ℙ^31.


There is a particular case when we know that the minors of a single flattening are enough to generate the ideal I(σ_s(X_𝐧)), namely the unbalanced case we already met in Theorem <ref>, for which we have the following result.


Let X = X_𝐧⊂ ^M with M=Π _i=1 ^t ( n_i +1)-1; let N=Π _i=1 ^t-1 ( n_i +1)-1; and let Y_(N,n_t) be the Segre embedding of ℙ^N×ℙ^n_t into ℙ^M. Assume n_t > N - ∑ _i=1^t-1n_i +1.
Then, for:

    N - ∑ _i=1 ^t-1 n_i +1 ≤ s ≤min{ n_t,N},

we have that σ_s(X) = σ_s(Y) ≠ℙ^M, and its ideal is generated by the (s+1)× (s + 1) minors of an (n_t + 1) × (N + 1) matrix of indeterminates, i.e., the flattening of the generic tensor with respect to the splitting {1,…,t-1}∪{t}.


We can notice that in the case above, X is defective for N - ∑ _i=1 ^t-1 n_i +1 < s; see Theorem <ref>, while when equality holds, σ_s(X) has the expected dimension. Moreover, in the cases covered by the theorem, σ_s(X) is arithmetically Cohen–Macaulay, and a minimal free resolution of its defining ideal is given by the Eagon–Northcott complex.



  §.§.§ Veronese Varieties

Now, let us consider the case of Veronese varieties. One case for which we have a rather complete information about the ideals of their higher secant varieties is the family of rational normal curves, i.e., the Veronese embeddings of ^1. In this case, the ideals in question are classically known; in particular, the ideal of σ_s(X_1,d) is generated by the (s+1)× (s+1) minors of catalecticant matrices associated with generic binary form of degree d, whose coefficients corresponds to the coordinates of the ambient coordinates. Moreover, we also know the entire minimal free resolution of these ideals, again given by the Eagon–Northcott complex.

Since the space of quadrics can be associated with the space of symmetric matrices, a similar analysis as the one done for the Segre varieties with two factors can be done in the case d = 2. In particular, the defining ideals for the higher secant varieties of the quadratic Veronese embeddings of ^n, i.e., of σ_s(X_n,2), are defined by the (s+1)× (s+1) minors of the generic symmetric matrix of size (n+1)×(n+1).

For any n,d, the ideal of σ_2(X_n,d) is considered in <cit.>, where it is proven that it is
generated by the 3× 3 minors of the first two catalecticant matrices of the generic polynomial of degree d in n+1 variables; for the ideal of the 3× 3 minors, see also <cit.>.

In general for all σ_s(X_n,d)'s, these kinds of equations, given by (s+1)× (s+1) minors of catalecticant matrices (<cit.>), are known, but in most of the cases, they are not enough to generate the whole ideal.


Notice that those catalecticant matrices can also be viewed this way (see <cit.>): consider a generic symmetric tensor (whose entries are indeterminates) T; perform a flattening of T as we just did for generic tensors and Segre varieties; erase from the matrix that is thus obtained all the repeated rows or columns. What you get is a generic catalecticant matrix, and all of them are obtained in this way, i.e., those equations are the same as you get for generic tensors, symmetrized.


Only in a few cases, our knowledge about the equations of secant varieties of Veronese varieties is complete; see for example <cit.>. All recent approaches employ representation theory and the definition of Young flattenings. We borrow the following list of known results from <cit.>.




	σ_s(X_n,2)     size s+1 minors generic symmetric matrix     ideal     classical 

	σ_s(X_1,d)     size s+1 minors of any generic catalecticant     ideal     classical 

	σ_2(X_n,d)     size 3 minors of     ideal     <cit.> 

	      generic (1,d-1) and (2,d-2)-catalecticants         

	σ_3(X_n,3)     Aronhold equation + size 4 minors of     ideal     Aronhold (n=2) <cit.> 

	      generic (1,2)-catalecticant         <cit.> 

	σ_3(X_n,d)     size 4 minors of     scheme     <cit.> (n = 2, d = 4) 

	  (d ≥ 4)     generic (1,3) and (2,2)-catalecticant         <cit.> 

	σ_4(X_2,d)     size 5 minors of     scheme     <cit.> (d = 4) 

	      generic (⌊d/2⌋,⌈d/2⌉)-catalecticant         <cit.> 

	σ_5(X_2,d)     size 6 minors of     scheme     Clebsch (d=4) <cit.>

	(d ≥ 6, d = 4)       generic (⌊d/2⌋,⌈d/2⌉)-catalecticant         <cit.> 

	σ_s(X_2,5)     size 2s+2 sub-Pfaffians of     irred.
	comp.
	    <cit.> 

	s ≤ 5       generic Young ((31),(31))-flattening     

	σ_6(X_2,5)     size 14 sub-Pfaffians of     scheme     <cit.> 

	s ≤ 5       generic Young ((31),(31))-flattening     
	
	σ_6(X_2,d)     size 7 minors of     scheme     <cit.> 

	      generic (⌊d/2⌋,⌈d/2⌉)-catalecticant     

	σ_7(X_2,6)     symmetric flattenings + Young flattenings     irred. comp.     <cit.> 

	σ_8(X_2,6)     symmetric flattenings + Young flattenings     irred. comp.     <cit.> 

	σ_9(X_2,6)     determinant of generic (3,3)-catalecticant     ideal     classical 

	σ_s(X_2,7)     size (2s+2) sub-Pfaffians of     irred. comp.     <cit.> 

	 (s ≤ 10)       generic ((4,1),(4,1))-Young flattening         

	σ_s(X_2,2m)     rank of (a,d-a)-catalecticant ≤min{s,a+2  2}     scheme     <cit.> 

	 (s ≤m+1  2)       for 1 ≤ a ≤ m, open and closed         

		



	σ_s(X_2,2m+1)     rank of (a,d-a)-catalecticant ≤min{s,a+2  2}     scheme     <cit.> 

	 (s ≤m+1  2+1)       for 1 ≤ a ≤ m, open and closed         

	σ_s(X_n,2m)     size s+1 minors of     irred. component     <cit.> 

	 (s ≤m+n-1  n)       generic (m,m)-catalecticant         
	
	σ_s(X_2,2m+1)     size n ⌊ n/2 ⌋j+1 minors of a Young flattening     irred. comp.     <cit.> 

	  (s ≤m+n  n)             







 Note that the knowledge of equations that define the σ_s(X_n,d)'s, also just
set-theoretically, would give the possibility to compute the symmetric border rank for any tensor in S^dV.


For the sake of completeness, we mention that equations for secant varieties in other cases can be found in <cit.> (Grassmannian and other homogeneous varieties), in <cit.> (Segre–Veronese varieties and Del Pezzo surfaces) and in <cit.> (Veronese re-embeddings of varieties). 



 §.§ The Real World

For many applications, it is very interesting to study tensor decompositions over the real numbers.

The first thing to observe here is that since  is not algebraically closed, the geometric picture is much more different. In particular, a first difference is in the definition of secant varieties, where, instead of considering the closure in the Zariski topology, we need to consider the Euclidean topology. In this way, we have that open sets are no longer dense, and there is not a definition of “generic rank”: if X is variety in ^N_, the set U_r(X) = {P ∈_^N  | _X(P) = r} might be non-empty interior for several values of r. Such values are called the typical ranks of X. 

It is known that the minimal typical (real) rank of X coincides with the generic (complex) rank of the complexification X ⊗; see <cit.>.

The kind of techniques that are used to treat this problem are sometimes very different from what we have seen in the case of algebraically-closed fields. 


Now, we want to overview the few known cases on real symmetric-ranks. 

The typical ranks of binary forms are completely known. Comon and Ottaviani conjectured in <cit.> that typical ranks take all values between ⌊d+2/2⌋ and d. The conjecture was proven by Blekherman in <cit.>. 


In <cit.>, the authors showed that any value between the minimal and the maximal typical rank is also a typical rank. Regarding real symmetric-ranks, they proved that: the typical real rank of ternary cubics is four; the typical ranks of quaternary cubics are only five and six; and they gave bounds on typical ranks of ternary quartics and quintics. 


Another family of symmetric tensors for which we have some results on real ranks are monomials. First of all, note that the apolarity lemma (Lemma <ref>) can still be employed, by making all algebraic computations over the complex number, but then looking for ideals of reduced points apolar to the given homogeneous polynomial and having only real coefficients. This was the method used in <cit.> to compute the real rank of binary monomials. 

Indeed, if M = x_0^α_1x_1^α_1, then, as we have already seen, M^⊥ = (y_0^α_0+1,y_1^α_1+1). Now, Waring decompositions of M are in one-to-one relation with reduced sets of points (which are principal ideals since we are in ^1), whose ideal is contained in M^⊥. Now, if we only look for sets of points that are also completely real, then we want to understand for which degree d it is possible to find suitable polynomials H_0 and H_1 such that G = y_0^α_0+1H_0 + y_1^α_1+1H_1 is of degree d and have only distinct real roots.

The authors observe the following two elementary facts, which hold for any univariate g(y) = c_dy^d+c_d-1y^d-1+…+c_1y+c_0, as a consequence of the classic Descartes' rule of signs:

	
  * if c_i = c_i-1 = 0, for some i = 1,…,d, then f does not have d real distinct roots; see <cit.> (Lemma 4.1);
	
  * for any j = 1,…,d-1, there exists c_i's such that f has d real distinct roots and c_j = 0; see <cit.> (Lemma 4.2).

As a consequence of this, we obtain that:

	
  * if d < α_0+α_1, then G (or rather, its dehomogenization) has two consecutive null coefficients; hence, it cannot have d real distinct roots;
	
  * if d = α_0+α_1, then only the coefficient corresponding to y_0^α_0y_1^α_1 of G (or rather its dehomogenization) is equal to zero; hence, it is possible to find H_0 and H_1 such that G has d real distinct roots.

Therefore, we get the following result.

If α_0,α_1 are not negative integers, then 
_ sym^ℝ(x_0^α_0x_1^α_1) =α_0+α_1.

Note that, comparing the latter result with Theorem <ref>, we can see that for binary monomials, the real and the complex rank coincide if and only if the least exponent is one. However, this is true in full generality, as shown in <cit.>.

Let M = x_0^α_0⋯ x_n^α_n be a degree d
 monomial with α_0 = min_i{α_i}. Then,

    _ sym^ℝ(M) = _ sym^ℂ(M)    if and only if   α_0 = 1.


Note that the real rank of monomials is not known in general, and as far as we know, the first unknown case is the monomial x_0^2x_1^2x_2^2, whose real rank is bounded by 11 ≤_ sym^ℝ(x_0^2x_1^2x_2^2) ≤ 13; here, the upper bound is given by <cit.>, and the lower bound is given by <cit.>.

We conclude with a result on real ranks of reducible real cubics, which gives a (partial) real counterpart to Theorem <ref>.

If F ∈[x_0,…,x_n] is a reducible cubic form essentially involving n+1 variables, then one and only one of the following holds:


  * F is equivalent to x_0(∑_i=1^nϵ_i x^2_i), where ϵ_i ∈{-1, +1}, for 1≤ i ≤ n, and:

    2n ≤_ sym^(F) ≤ 2n + 1.

Moreover, if ∑_i ϵ_i = 0, then _ sym^(F) = 2n.

  * F is equivalent to x_0(∑_i=0^n ϵ_ix_i^2), where ϵ_i ∈{-1, +1}, for 1≤ i ≤ n, and:

    2n ≤_ sym^(F) ≤ 2n + 1.

Moreover, if ϵ_0 = … = ϵ_n, then _ sym^(F) = 2n. If ϵ_0 ≠ϵ_1 and ϵ_1=…=ϵ_n,
then _ sym^(F) = 2n + 1.

  * F is equivalent to (α x_0 + x_p)(∑_i=0^n ϵ_i x_i^2), for α≠ 0, where ϵ_0=…=ϵ_p-1=1 and ϵ_p=…=ϵ_n = -1 for 1 ≤ p ≤ n, and: 
    2n ≤_ sym^(F) ≤ 2n + 3.

Moreover, if α = -1 or α = 1, then 2n + 1 ≤_ sym^(F) ≤ 2n + 3.




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