The non-Lefschetz locus of conics
Emanuela Marangone

	Mitigating Automotive Radar Interference using Onboard Intelligent Reflective Surface

    Shree Prasad Maruthi^*, Karrthik G. K.^**, Vijaya Krishna A.^†, Mahbub Hassan^, Jinhong Yuan^
		^*Dept. of Electrical and Electronics Engineering, WILP Division, BITS Pilani, India 

			^**Charcoal Technologies, Bengaluru, India, ^†Bosch Global Software Ltd., Bengaluru, India 

			^ School of Computer Science and Engineering, University of New South Wales, Sydney, Australia

			^ School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, Australia

			Email: shreeprasadm@gmail.com, karrthik.gk23@gmail.com, A.VijayaKrishna@in.bosch.com, mahbub.hassan@unsw.edu.au,
 j.yuan@unsw.edu.au
		

    
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A graded Artinian algebra A  has the Weak Lefschetz Property if there exists a linear form ℓ such that the multiplication map by ℓ:[A]_i→ [A]_i+1 has maximum rank in every degree. The linear forms satisfying this property form a Zariski-open set; its complement is called the non-Lefschetz locus of A.


In this paper, we investigate analogous questions for degree-two forms rather than lines. We prove that any complete intersection A=k[x_1,x_2,x_3]/(f_1,f_2,f_3), with k=0, has the Strong Lefschetz Property at range 2, i.e. there exists a linear form ℓ∈ [R]_1, such that the multiplication map ×ℓ^2: [M]_i→ [M]_i+2 has maximum rank in each degree.

Then we focus on the forms of degree 2 such that × C: [A]_i→ [A]_i+2 fails to have maximum rank in some degree i. The main result shows that the non-Lefschetz locus of conics for a general complete intersection A=k[x_1,x_2,x_3]/(f_1,f_2,f_3) has the expected codimension as a subscheme of ^5. The hypothesis of generality is necessary. We include examples of monomial complete intersections in which the non-Lefschetz locus of conics has different codimension. 

To extend a similar result to the first cohomology modules of rank 2 vector bundles over ^2, we explore the connection between non-Lefschetz conics and jumping conics. The non-Lefschetz locus of conics is a subset of the jumping conics. Unlike the case of the lines, this can be proper when  is semistable with first Chern class even. 
	







§ INTRODUCTION


A graded Artinan algebra A  has the Weak Lefschetz Property (WLP) if the multiplication map by a linear form ℓ has maximum rank in every degree. If the multiplication map by any power of a general linear form ℓ has maximum rank as well, we say that A has the Strong Lefschetz Property (SLP).
This last property also implies the Maximum rank property (MRP) i.e, for any d, the multiplication by a general form of degree d always has maximum rank.
In this paper, we study the forms of degree 2 such that × C: [A]_i→ [A]_i+2 fails to have maximum rank in some degree i. 

The most famous result in the study of the Lesfchetz Property states that every Artinian monomial complete intersection over a field k of characteristic zero has the SLP <cit.>, <cit.> <cit.>. As a consequence of this theorem, a general complete intersection k[x_1,…,x_n]/(f_1,…, f_n) with fixed generator degrees has the SLP. However, it is an open question to determine whether every complete intersection has the SLP or even the WLP.  The most important result in this direction proves the Weak Lefschetz Property for n=3. To prove this theorem, Harima–Migliore–Nagel–Watanabe <cit.> applied the Grauert-Mülich Theorem to the syzygy bundle, in this case, a locally free sheaf of rank 2 over ^2. 

The study of the non-Lefschetz locus, defined as the set of linear forms that fail to have maximum rank in some degree,  started with the work of Boij–Migliore–Miró-Roig–Nagel <cit.>. In  <cit.>, the authors conjecture that the non-Lefschetz locus of a general complete intersection has the expected codimension. This conjecture has been proven for general complete intersections if n=3 <cit.>. 
Failla–Flores–Peterson <cit.> generalize the study of the Lefschetz Property to first cohomology module _*^1(^2,) of any vector bundle  of rank 2 over ^2 and prove that these finite-length modules have the WLP.  The non-Lefschetz locus of _*^1(^2,) is exactly the set of jumping lines, and it has the expected codimension under the assumption that  is general <cit.>.

We investigate analogous questions for higher-degree forms rather than lines. This problem is connected with the SLP at range 2 and with the Maximum Rank Property (MRP). In analogy with <cit.>, we endow  
 
    𝒞_A,i={C∈[R]_2  | × C: [A]_i→ [A]_i+2 does not have max rank}
 with a scheme structure given by the ideal I(𝒞_A,i) of maximal minors of a suitable h_i+2× h_i  matrix of linear forms. 𝒞_A is the subscheme of ^n+12-1 defined by the ideal I(ℒ_A)=⋂ I(ℒ_A,i).
In this paper, we will mainly focus on the case n=3. In this case, we refer to _A as the non-Lefschetz locus of conics of A.
The main result provides a version of <cit.> for conics.
[<ref>]
	The non-Lefschetz locus of conics for a general complete intersection of height 3 has the expected codimension and degree as a subscheme of ^5.




This paper is organized as follows.
In 2, following what has been done for lines in <cit.>, we define the non-Lefschetz locus for forms of degree 2, for any Artinian algebra A over R=k[x_1,…, x_n], as a subscheme ^n+12-1. We will prove that
[<ref>]
Let A be a Gorenstein Artinian algebra of socle degree e. If A has the WLP, then the non-Lefschetz locus of forms of degree 2 is defined by the ideal in the middle degree. 



In 3, we specialize to the case in  3 variables. In this setting, we refer to 𝒞_A with its structure as a subscheme of ^5 as the non-Lefschetz locus of conics of A.
Our goal is to study the non-Lefschetz locus of conics for the first cohomology modules _*^1(^2,) of a rank 2 vector bundles , as a generalization of the case of complete intersections  R/(f_1,f_2,f_3). Therefore in this section, we provide some background on vector bundles over ^2.

In 4, we give the following result.
 [<ref>]
	Any complete intersection A=k[x_1,x_2,x_3]/(f_1,f_2,f_3) has the Strong Lefschetz Property at range 2.
   
   Moreover, the lines ℓ for which the multiplication map by ℓ^2 does not have maximal rank form a hypersurface in the space of linear forms. 
 The same is true as well for any first cohomology module of a rank 2 vector bundle _*^1(^2,) over ^2 (Proposition <ref>).
As a consequence the non-Lefschetz locus of conics of M=_*^1(^2,) has positive codimension in ^5. 


In 5, we investigate the connection between non-Lefschetz conics and the jumping conics. The notion of jumping conics was initially introduced by <cit.> for semistable vector bundles of rank 2 over ^2.  We first extend the definition to include the case when  is unstable. It follows that whenever  is unstable or  c_1() is odd, the jumping conics are exactly the non-Lefschetz conics. In such cases, the non-Lefschetz locus of conics _M is a hypersurface in ^5. This does not hold when  is semistable with c_1() even: while every non-Lefschetz conic is indeed a jumping conic, the reverse is not necessarily true.
[<ref>]
	Let  be a semistable, normalized vector bundle with c_1()=0.

 A smooth conic C is a non-Lefschetz conic if and only if it is a jumping conic such that _| C≅_^1(a)⊕_^1(-a) with a>1.

In 5.1, we compute the expected codimension of _M. For a semistable  with even first Chern class, the non-Lefschetz locus of conics is expected to have codimension 2 if  is semistable but not stable, and 3 if  is stable. We conjecture that such dimension is achieved for  general. 

In 6, we resolve the question about the codimension of the non-Lefschetz locus of conics for a general complete intersection, proving the conjecture for this particular case.
[<ref>]
	Let A=R/(f_1,f_2,f_3) be a general complete intersection of type (d_1,d_2,d_3), and socle degree e. The non-Lefschetz locus of conics has the expected codimension in ^5:
	
    𝒞_A= 1    if  e  is even or  d_3≥ d_1+d_2+2;
    
    		2    if  d_3=d_1+d_2
    
    		3    if  e  odd and d_3≤ d_1+d_2-2.


The most interesting case is when the socle degree e is odd and d_3≤ d_1+d_2. For this case, in 6.1 we construct an explicit Gorenstein algebra R/J  with the desired Hilbert function for which the non-Lefschetz locus of conics has expected codimension and then invoke semicontinuity.
 In 6.2, for the case d_3< d_1+d_2-2,  we also prove that the set of conics in _A that do not vanish at any of the points of the zero-dimensional scheme defined by the ideal (f_1,f_2), has codimension  3 in ^5.

The hypothesis of generality is necessary whenever the socle degree e is odd and d_3≤ d_1+d_2. In fact, in 7  we construct examples of monomial complete intersection A=R/(x_1^d_1,x_2^d_2,x_3^d_3) 
with d_3≤ d_1+d_2-2, where _A has codimension 1, 2, or 3.
Moreover, we show that for every monomial complete intersection with d_3=d_1+d_2, _A is always a hypersurface even if, in this case, the expected codimension is 2.

In 8, we study the non-Lefschetz locus of conics for general height 3 Gorenstein algebras with fixed Hilbert function. As in the case of lines studied in <cit.>, to have the expected codimension we need an extra condition on the g-vector (1,2,g_2,…,g_⌊e/2⌋), defined as the positive part of the first difference of the Hilbert function.
[<ref>]
	Let (1,3,h_2,…,h_e) be an SI-sequence such that its g-vector is of decreasing type. 
	Then for a general Gorenstein algebra with Hilbert function (1,3,h_2,…,h_e) the non-Lefschetz locus of conics has expected codimension in ^5.
 
 Without the condition on the first difference, the expected codimension might not be achieved. Given a SI-sequence (1,3,h_2,…,h_e) with g-vector not of decreasing type, the non-Lefschetz locus of conics of a general Gorenstein algebra with Hilbert function (1,3,h_2,…,h_e), always has codimention 1. 

 








§ THE NON-LEFSCHETZ LOCUS OF FORMS OF DEGREE 2


In this section, we want to define the non-Lefschetz locus for forms of degree 2 for any Artinian algebra, following what has been done for linear forms in <cit.>.
We will first recall the definitions of Lefschetz Properties and non-Lefschetz locus.  

Let k be an algebraic closed field of characteristic 0, and let R=k[x_1,…,x_n]  be the standard graded polynomial ring in n variables.


Let A=R/I a graded Artinian algebra of socle degree e, A=⊕_i=0^e [A]_i. We denote with (h_0,…, h_e) the Hilbert function of A, i.e. h_i= ([A]_i) for each 0≤ i≤ e. 

A has the Weak Lefschetz Property (WLP) if there exists a linear element ℓ in R such that the multiplication map ×ℓ: [A]_i→ [A]_i+1 has maximal rank for every  degree i, i.e. it is always either surjective or injective. Such an element is called a (Weak) Lefschetz element.

That set of Lefschetz elements form a Zariski-open set, and its complement is called the non-Lefschetz locus of A and denoted with _A. 

	We say that A has the Strong Lefschetz Property (SLP) if there is a linear element ℓ such that the multiplication map for any power of ℓ, ×ℓ^d:[A]_i→ [A]_i+d, has maximal rank in each degree i.


The SLP also implies the Maximum Rank Property:

A has the Maximal Rank Property (MRP) if for each d the multiplication map for any general form of degree d, f∈[R]_d, has maximal rank in each degree.

If we fix d we also have

	We say that A has the Strong Lefschetz Property (SLP) at range d if there is a linear element ℓ such that the multiplication map ×ℓ^d:[A]_i→ [A]_i+d, has maximal rank in each degree i.


In this paper, we want to focus only on the case d=2, and study the forms C∈ [R]_2 such that there exists a degree i for which the multiplication map × C: [A]_i→ [A]_i+2 is neither surjective nor injective.

	
		For the Artinian algebra A of socle degree e, we define the non-Lefschetz locus of forms of degree 2 as
		
    _A=⋃_i=0^e-2_A,i
 where 
            
    _A,i={C∈ [R]_2:  × C: [A]_i→ [A]_i+2 does not have maximum rank }.

	
 
 
We want to study the non-Lefschetz locus of forms of degree 2 as a subscheme of ^N-1, for N=n+12, in a similar way to what has been done for the non-Lefschetz locus _A in <cit.>. 

For each i  we have a map 

    ϕ :[R]_2    →_k([A]_i,[A]_i+2)
    
    	C   ↦× C: [A]_i → [A]_i+2.

Given a choice of basis for [A]_i and [A]_i+2 as k-vector spaces, ϕ(C)∈_k([A]_i,[A]_i+2) is represented by a h_i+2× h_i matrix whose entries linearly depend on the coefficients of the C. Then C∈_i,A if and only if the ϕ(C) < min{h_i,h_i+2}, i.e. if one of the maximal minors of ϕ(C) is zero, and this does not depend on the choice of the basis.
 
Following this idea, we define S=k[a_1,…,a_N] to be the coordinate ring associated with ^N-1. We can think of the variables a_1,…,a_N as the coefficients of a conic 
    C=a_1x_1^2+a_2x_1x_2+…+a_Nx_n^2.


The multiplication map × C on S⊗_k A give us  a map 

    × C : S⊗_k [A]_i → S⊗_k [A]_i+2

that, chosen a basis for [A]_i and [A]_i+2, can be represented by a h_i+2× h_i matrix of linear forms in S. We call this matrix  B_i.
  

	_A,i⊆^N-1 is scheme-theoretically defined by the ideal I(_A,i) of maximal minors of the matrix B_i. The non-Lefschetz locus of forms of degree 2, _A, is defined as a subscheme of ^N-1 by the homogeneous ideal I(_A)=⋂_0≤ i≤ e-2I(_A,i).


    This definition can be applied to forms of higher degrees as well; in this paper, we analyze just the case of degree 2.

The expected codimension  of ℒ_A,i is |h_i+2-h_i|+1 and if this codimension is achieved then ℒ_I,i=h_i+2h_i-1  by <cit.>. 

The non-Lefschetz locus _A of forms of degree-two forms is defined a priori as a union of determinantal schemes. However, to get a clearer understanding (and in particular to facilitate the computation of the expected codimension) we need to understand which condition we need in order to get a containment between the ideals I(_A,i) and determine that _A is “concentrated” in one degree.


Similarly to what is done for lines in <cit.>, we have the following proposition.
 
		If h_i≤ h_i+2≤ h_i+4 and there is no socle in degree i and i+1, then I(_A,i+2)⊆ I(_A,i).
 

The proof is divided into three parts. First, we will show that we can reduce to the case h_i+2=h_i+4. In the second step, we take care of the case when one of the ideals is 0. Finally, in the third part, we assume h_i+2=h_i+4 and I(_A,i+2)≠0≠ I(_A,i) and prove the desired inclusion.

Step 1. Here we reduce to the case when h_i+2=h_i+4.

Fix bases for [A]_i+2 and [A]_i+4. Recall that I(_A,i+2) is the ideal of maximal minors of the h_i+4× h_i+2 matrix B_i+2, given by the map

    S⊗_k [A]_i+2B_i+2→ S⊗_k [A]_i+4.

Denote with {v_1,…, v_h_i+4} the chosen basis for [A]_i+4. For each  J⊆{1,…, h_i+4} consisting of h_i+2 elements, we define  A^J to be the algebra obtained by quotienting A by the ideal generated by the forms of degree i+4 indexed by elements not in J:

    A^J=A/(v_j  |  j∉ J).


Since we quotient by an ideal generated in degree i+4, [A^J]_j=[A]_j for every j<i+4. In particular, [A^J]_i=[A]_i and [A^J]_i+1=[A]_i+1, so I(_A^J,i)=I(_A,i). 
Consider the diagram 
 
    S⊗_k [A]_i+2rB_i+2[equal]d    S⊗_k [A]_i+4[two heads]d
    
    S⊗_k [A^J]_i+2rB^J   S⊗_k [A^J]_i+4.

Under these identifications, the lower map is represented by the submatrix B^J, obtained from B_i+2 by considering just the rows indexed by elements of J. In fact, B^J is a maximal square submatrix of B_i+2, and all the maximum submatrices are of this form, for some J⊆{1,…, h_i+4} with  |J|=h_i+2.
Since I(_A^J,i+2) is defined to be the ideal of maximal minors of B^J, we have ( B^J)=I(_A^J,i+2).

In order to prove that I(_A,i+2)⊆ I(_A,i), we need to show that each maximal minor of the  B_i+2 is in the ideal I(_A,i), equivalently that for each J, ( B^J)⊆ I(_A,i). By the equalities proved above this is the same as to prove that  I(_A^J,i+2)⊆ I(_A^J,i), for each  J⊆{1,…, h_i+4} subset of cardinality h_i+2.

If we prove the proposition for each  Artinian algebra with  h_i≤ h_i+2=h_i+4 and no socle in degree i and i+1, then the statement will follow.



Step 2. Consider the case when one of the ideals is zero.
If I(_A,i+2)=0, then there is nothing to prove. When I(_A,i)=0, it is enough to show that _A,i+2=^N-1. 
Assume by contradiction that ∃ C∈ [R]_2 such that the multiplication map × C:[A]_i+2→ [A]_i+4 is injective. Since we assume I(_A,i)=0, _A,i=^N-1and, in particular, there exists a f∈ [A]_i non-zero with Cf=0. A does not have socle in degree i or i+1, so ∃ℓ, ℓ' ∈ [R]_1 such that fℓℓ'≠ 0 in [A]_i+2. But C fℓℓ'=0 contradicting injectivity of × C:[A]_i+2→ [A]_i+4.


Step 3. Let us assume h_i+4=h_i+2 and _A,i+2≠^N-1≠_A,i.
Then we can choose a conic C∈[R]_2, such that × C:[A]_i+2→ [A]_i+4 is bijective, and so × C:[A]_i→ [A]_i+2 injective, by the argument in Step 2. We have the diagram

    S⊗_k [A]_irB_i[hook, swap]d× C      S⊗_k [A]_i+2d270∼  × C
    
    S⊗_k [A]_i+2rB_i+2    S⊗_k [A]_i+4.

Since C is fixed the vertical maps are the identity on S. Hence, chosen appropriate basis for [A]_i+2 and [A]_i+4, B_i is a submatrix of B_i+2.
We assume h_i+2=h_i+4, so B_i+2 is a square matrix, and B_i is a submatrix with the same number of rows h_i+2 as B_i+2.  Then,  B_i+2 can be written as a linear combination of the maximal minors of B_i, and

    I(C_A,i+2)=( B_t+2)⊆(max minors of  B_i)=I(C_A,i)

as desired.









Proposition <ref> holds if we consider finite length module M instead of Artinian algebras. Moreover, we also have a dual statement: if h_i≥ h_i+2≥ h_i+4 and there are no new generators in degree in degree i+3 and i+4, i.e. (x_1,…,x_n)[M]_i+2=[M]_i+3 and (x_1,…,x_n)[M]_i+3=[M]_i+4, then I(_M,i+2)⊇ I(_M,i).

 

	Let A be a Gorenstein Artinian algebra of socle decree e. If A has the Weak Lefschetz Property, then the non-Lefschetz locus of forms of degree 2 is defined as subscheme of ^N-1 by the ideal in the middle degree 
    I(_A)= I(_A,⌊e/2⌋-1).



	The WLP guarantees the unimodality of the Hilbert function. Then applying Proposition <ref> we
   have that 
  
    e  even:    
       ⋯≤ h_e/2-4≤ h_e/2-2≤ h_e/2   ⇒   
       ⋯⊇ I(_A,e/2-4)⊇ I(C_A,e/2-2) 
             ⋯≤ h_e/2-3≤ h_e/2-1= h_e/2+1   ⇒   
       ⋯⊇ I(_A,e/2-3)⊇ I(_A,e/2-1) 
       e  odd:    
       ⋯≤ h_e-1/2-4≤ h_e-1/2-2≤ h_e-1/2   ⇒   
       ⋯⊇ I(_A,e-1/2-4)⊇ I(_A,e-1/2-2)   
             ⋯≤ h_e-1/2-3≤ h_e-1/2-1≤ h_e+1/2   ⇒   
       ⋯⊇ I(_A,e-1/2-3)⊇ I(_A,e-1/2-1)

where  e is the socle degree of A.

 Since A is Gorenstein, duality shows that, for each i≥ 0, I(_A,⌊e/2⌋-i-2)=I(_A,⌊e+1/2⌋+i)
  and so we have
  
    I(_A)=⋂_i=0^e-2I(_A,i)=I(_A,⌊e/2⌋-1)∩ I(_A,⌊e/2⌋-2).

  
 To prove 
 I(_A)= I(_A,⌊e/2⌋-1)
 we will show
 
    I(_⌊e/2⌋-1)⊆ I(_⌊e/2⌋-2)

 using the assumption that A has the WLP. We will divide the proof depending on the parity of the socle degree e.
 
 Case e odd. Fix a Weak Lefschetz element ℓ∈ [R]_1.
 Then ×ℓ:[A]_e-1/2-2→ [A]_e-1/2-1 is injective and ×ℓ:[A]_e-1/2→ [A]_e+1/2 is bijective (h_e-1/2=h_e+1/2). We have the diagram
 
    S⊗_k [A]_e-1/2-2rB_e-1/2-2[hook, swap]d×ℓ     S⊗_k [A]_e-1/2-1d270∼  ×ℓ
    
    S⊗_k [A]_e-1/2-1rB_e-1/2-1   → S⊗_k [A]_e+1/2.

Then, choosing  appropriate bases, we can see B_e-1/2-2 as a submatrix of B_e-1/2-1.
Since h_e-1/2=h_e+1/2, B_e-1/2-1 and B_e-1/2-2 have the same number of rows, that is greater than or equal to the number of columns of B_e-1/2-1. This implies that each maximal minor of B_e-1/2-1 can be seen as a linear combination of maximal minors of B_e-1/2-2. Hence,

    I(_A,e-1/2-1)=(max minors of  B_e-1/2-1)⊆(max minors of  B_e-1/2-2)=I(_A,e-1/2-2).




 Case e even. By duality I(C_A,e/2-2)=I(C_A,e/2). So it is enough to show 
 
    I(_e/2-1)⊆ I(_e/2).

 Since we assume A has the WLP we can choose a linear form  ℓ∈ [R]_1 such that the map
 ×ℓ:[A]_e/2-1→ [A]_e/2 is injective and ×ℓ:[A]_e/2→ [A]_e/2+1 is surjective. Then we have the commutative diagram
  
    S⊗_k [A]_e/2-1rB_e/2-1[hook, swap]d×ℓ     S⊗_k [A]_e/2+1d270∼  ×ℓ
    
    S⊗_k [A]_e/2rB_e/2    S⊗_k [A]_e/2+2

and, with appropriate choices of bases, the diagonal map 
can be represented by a matrix B of linear forms in S such that B is a submatrix of B_e/2-1 and B_e/2 is a submatrix of B:
 
    S⊗_k [A]_e/2-1rB_e/2-1[hook, swap]d×ℓ drB    S⊗_k [A]_e/2+1d270∼  ×ℓ
    
    S⊗_k [A]_e/2rB_e/2    S⊗_k [A]_e/2+2.

First we notice that B_e/2-1 is a square sub-matrix of B, with the same number of columns as B, so

    I(_A,e/2-1)=( B_e/2-1)⊆ (max minors of  B).

On the other side B is a submatrix of B_e/2, with the same number of rows, that is smaller or equal to h_e/2, the number of columns of B_e/2. Therefore each maximal minor of B is a maximal minor of B_e/2 as well.

    I(_A,e/2-1)⊆ (max minors of  B)⊆(max minors of  B_e/2)=I(_A,e/2).



	The hypothesis of the Weak Lefschetz property in Theorem <ref> is necessary. Otherwise, we can only conclude that 
	
    I(_A)= I(_A,⌊e/2⌋-1)∩ I(_A,⌊e/2⌋-2).

	











§ FIRST COHOMOLOGY MODULE OF A RANK 2 VECTOR BUNDLE OVER ^2


In this section, we will focus on the case of three variables. It is easy to see that all the definitions introduced in the previous section for Artinian algebras can be extended to finite-length modules over the polynomial ring R.
For any finite length graded module M over R=k[x_1,x_2,x_3], we say that a form C∈ [R]_2 is a Lefschetz Conic  if the multiplication map × C: [M]_i→ [M]_i+2 has maximum rank in each degree, and we refer to _M (Definition <ref>) with its structure as subscheme of ^5 as the non-Lefschetz locus of conics of M.

Our goal is to study the non-Lefschetz locus for a height three complete intersection A=k[x_1,x_2,x_3]/(f_1,f_2,f_3), and as a generalization any finite length module M that is the cokernel of a
graded map 
    φ:⊕_i=1^n+2R(-a_i)→⊕_i=1^n R(-b_i).


In <cit.>, the authors showed that  M=φ is isomorphic to the first cohomology module of a rank 2 vector bundle  over ^2: 
    M≅_*^1(^2, ):=⊕_t∈^1(^2, (t)).
 Note that if n=1 and b_1=0 then the cokernel is a complete intersection R/(f_1,f_2,f_3) with f_i= a_i.
These two descriptions are equivalent since any first cohomology module of a rank 2 vector bundle over ^2, is of such form. This result is likely known to experts; however, since we were not able to find a proof in the literature, a proof is included for completeness.


    Let  be a vector bundle of rank 2 over ^2, and let M=_*^1(^2, ). Assume M≠ 0. Then there exists integers n,a_i,b_i such that we have a exact sequence
    
    0→ E →⊕_i=1^n+2R(-a_i)→⊕_i=1^n R(-b_i)→ M→ 0

    where Ẽ=. 


    Let  be a rank 2 vector bundle, and let M=_*^1(^2, ):=⊕_t∈^1(^2, (t)).
    M is a finite length graded module over R=[x_1,x_1,x_3]  <cit.>.
    We assumed M≠ 0, excluding the case where  splits.
    Let E:=Γ_*(^2, )=⊕_t∈^0(^2, (t)). 
    
    E is a finitely generated graded R-module and Ẽ≅ <cit.>.
    Using <cit.>, we have that ^2_m(E)=_*^1(^2, )=M≠ 0, hence E≤ 2.
    Since Γ_*(^2, )=Γ_*(^2, )=E by definition of E, the depth of E must be exacly 2 <cit.> and so its projective dimension is 1. 
    Then E has a free resolution of length 2, that we can sheafify and dualize to get the exact sequence of sheaves 
    
    0→→⊕_i=1^n+2𝒪_ℙ^2(a̅_i)→⊕_i=1^n𝒪_ℙ^2 (b̅_i) → 0

    for some integers a̅_i,b̅_i.
Since  is a rank 2 vector bundle over ^2, it follows that  
≅(d), where -d=c_1() is the first Chern Class of  <cit.>.
Then shifting by -d we obtain the exact sequence of sheaves 

    0→→⊕_i=1^n+2𝒪_ℙ^2(-a_i)→⊕_i=1^n𝒪_ℙ^2 (-b_i) → 0

where we define a_i=d-a̅_i, and b_i=d-b̅_i.
Applying the functor Γ_* we get the sequence

    
    0→ E→⊕_i=1^n+2R(-a_i)→⊕_i=1^n R(-b_i) →_*^1(^2, )→ 0

 as desired.  


Let   is a rank 2 vector bundle over ^2 and M=_*^1(^2, ) its first cohomology module.


    A conic C is a  Lefschetz conic for M=_*^1(^2, ) if and only if 
    × C: [M]_i ^1(^2, (i))→[M]_i+2^1(^2, (i+2))
 has maximum rank for each i. Since this property is independent of the shift, we can always assume  normalized, hence it has first Chern Class c_1()∈{0,-1} (<cit.>). With an abuse of notation we will label with × C the map ^1(^2, (i)) →^1(^2, (i+2)).


Before proceeding with the study of the non-Lefschetz locus of conic of M=_*^1(^2, ) we recall some results about vector bundles over projective space, that we will use in the following sections. 


[<cit.>]
Let  be rank r vector bundle over ^n. Let c_1() be its first Chern class, and μ()=c_1()/r the slope of .
We say that 

    
  *  is semistable if for any non-zero coherent subsheaf ℱ⊂ the slope satisfies μ(ℱ)≤μ();
    
  *  is stable if such equality is strict, i.e. for any non-zero coherent subsheaf ℱ⊂,  μ(ℱ)<μ();
    
  * E is unstable it is not semistable.



Since we restrict attention to the case of (normalized) rank 2 vector bundles,  we use the following equivalent classification as the defining property for stability.

    Let  be a normalized rank 2 vector bundle over ^n. Then

    
  *  is stable if and only if it has no global sections, i.e. ^0(^n, )=0;
     
  * if c_1()=-1, stability and semistability are equivalent;
     
  * if c_1()=0,  is semistable if and only if  ^0(^n, (-1))=0. 



[<cit.>]
   For a an unstable normalized rank 2 vector bundle  over ^n we can define the instability index of  as the largest integer k such that ^0(^n, (-k))≠0. 
As a consequence, for an unstable vector bundle 

    
  * k>0 if c_1()=0
    
  * k≥0 if c_1()=-1.


Recall that every vector bundle over ^1 splits by Grothendieck's Theorem <cit.>, hence, for a rank 2 vector bundle over ^n, _|ℓ splits for any line ℓ in ^n.
[<cit.>]
	Given a line ℓ the splitting type of  on ℓ is the couple (a,b) where _|ℓ≅𝒪_ℙ^1(a)⊕𝒪_ℙ^2(b).


[ Grauert-Mülich Theorem <cit.>]
Let  be a semistable normalized rank 2 vector bundle on ^n and let ℓ be a general line. Then
	
    _|ℓ≅𝒪_ℙ^1⊕𝒪_ℙ^1    if  c_1()=0;
    𝒪_ℙ^1(-1)⊕𝒪_ℙ^1    if  c_1()=-1.



We are mainly interested in rank 2 vector bundles on ^2. The Grauert-Mülich Theorem gives us the splitting type of  over a general line ℓ when is semistable. For  unstable we have:
[<cit.>]Let  an unstable normalized vector bundle of rank 2 over ^2, and let k be the instability index of . Then for a general line ℓ 
	
    _|ℓ≅𝒪_ℙ^1(-k)⊕𝒪_ℙ^1(k)    if  c_1()=0;
    𝒪_ℙ^1(-k-1)⊕𝒪_ℙ^1(k)     if  c_1()=-1.

 

	A jumping line is a linear element ℓ with splitting type different from the splitting type of a general line.


Finally, if  is a semistable rank 2 vector bundle over ^2 with first Chern class odd, we have the following definition:
[<cit.>]
   A line ℓ is called a splitting line of second type for  when ^0(^2,_| ℓ^2)≠0.

Hulek in <cit.> proved in this case the following results:
[<cit.>]
    Let  be a semistable rank 2 vector bundle over ^2 with c_1()=0. Then the jumping lines of second type form a curve of degree 2(c_2()-1).

[<cit.>]
    Let  be a general semistable rank 2 vector bundle over ^2 with c_1()=0 and second Chern class c_2(). Then there are c_2()2 jumping lines, and they correspond to the singular points of the curve formed by the jumping lines of second type of .


    We can talk about general vector bundle since the moduli space of the rank 2 vector bundles over ^2 with c_1=-1 fixed second Chern class c_2 is irreducible by <cit.>. Similarly, also the moduli space of the rank 2 vector bundles with c_1=0  and fixed second Chern class c_2 is irreducible (see <cit.>).








§ STRONG LEFSCHETZ PROPERTY AT RANGE 2


To study the non-Lefschetz locus of conics _M⊂^5 we first want to show that it has positive codimension. Equivalently, we will show that there exists a Lefschetz conic. In fact, we will prove that there exists a linear form ℓ∈ [R]_1 such that the multiplication map 
×ℓ^2: [M]_i→ [M]_i+2 has maximum rank in each degree.  


    Let M=_*^1(^2,) be the first cohomology module of , a rank 2 vector bundle over ^2.  Then the multiplication map ×ℓ^2: [M]_i→ [M]_i+2, for a general linear form ℓ∈ [R]_1, has maximum rank in each degree.
    Moreover, the set of lines ℓ that fail to have this property form a hypersurface in (^2)^*.


    Without loss of generality, we can assume  is normalized. A general line ℓ is a Lefschetz element for M by <cit.>. In fact, by <cit.> we have:

    
  * If  is semistable and c_1()=0, then ×ℓ: [M]_i-1→ [M]_i is injective for i<-1, bijective for i=-1, and surjective for i >-1;
    
  * If  is unstable with index of instability k and c_1()=0, then k>0 and ×ℓ is injective for i<-(k+1), bijective for -(k+1) ≤ i ≤ k-1, and surjective for i >k-1;
    
  * If  is unstable with index of instability k and c_1()=-1, then k≥ and ×ℓ is injective for i<-(k+1), bijective for -k-1≤ i≤ k, and surjective for i >k.

    
    Case 1:  is unstable or c_1()=0. In this case, the multiplication map in the middle degree is an isomorphism. 
Then the multiplication by ℓ^2:[M]_i→ [M]_i+2 must always have maximum rank for each i.
This also shows that if  is unstable or if the first Chern class is even, the lines ℓ for which ×ℓ^2: [M]_i→ [M]_i+2 fails to have maximum rank in some degree are exactly the non-Lefschetz elements of M. These are the jumping lines of  by <cit.>, which, in this case, form a hypersurface in ^2.

 Case 2:  is semistable with c_1()=-1.
In this case by <cit.> the multiplication map ×ℓ: [M]_i-1→ [M]_i by a general line ℓ is injective for i≤ -1 and surjective for i≥ 0. Therefore ×ℓ^2: [M]_i→ [M]_i+2, when ℓ is a general linear form, is injective for i<-2 and surjective for i≥ -1. So it is enough to show that there exists ℓ∈ [R]_1 such that 
    ×ℓ^2: [M]_-2_*^1(^2, (-2))→[M]_0_*^1(^2, )
 has maximum rank.
From the short exact sequence
    
    0→(-2)×ℓ^2→→_| ℓ^2→ 0

we get the long exact sequence 

    0→^0(^2,(-2))→^0(^2,)→^0(^2,_| ℓ^2)→^1(^2,(-2))×ℓ^2→^1(^2,)→
    →^1(^2,_| ℓ^2)→^2(^2,(-2))→^2(^2,)→^2(^2,_| ℓ^2)=0.

Since  is semistable and c_1()=-1, ^0(^2,)=0. Using duality 

    ^2(^2,(-2))≅^0(^2,(-1))≅^0(^2,)=0
 since =(1) for a rank 2 vector bundle with c_1()=-1.
Then  the sequence above becomes 

    0→^0(^2,_| ℓ^2)→^1(^2,(-2))×ℓ^2→^1(^2,)
    	→^1(^2,_| ℓ^2)→ 0.

It is sufficient that ^0(^2,_| ℓ^2)=0 to obtain that ×ℓ^2: [M]_-2→ [M]_0 is injective, and so ℓ^2 has maximum rank. 
This condition is also necessary. In fact, using duality again we have that ^1(^2,(-2))≅^1(^2,(-1))≅^1(^2,), hence ×ℓ^2: [M]_-2→ [M]_0 has maximum rank if and only if it is an isomorphism.

So we conclude that ×ℓ^2: [M]_i→ [M]_i+2 fails to have maximum rank if and only if ^0(^2,_| ℓ^2)≠0, i.e. if and only if ℓ is a jumping line of  of second type (Definition <ref>). By Theorem <ref> these form a hypersurface in (^2)^*.
 In particular, the complement is not empty, so for a general line ℓ the multiplication map ×ℓ^2:[M]_i→ [M]_i+2 has maximum rank in each degree.



  Any Artinian complete intersection A=R/(f_1,f_2,f_3),  has the Strong Lefschetz Property at range 2, i.e. there exists a linear form ℓ∈ [R]_1 such that multiplication map ×ℓ^2: [A]_i→ [A]_i+2 has maximum rank in each degree. 


    The first syzygy  bundle of A is a rank 2 vector bundle over ^2, and A≅_*^1(^2,), so we can apply Proposition <ref>.



    The proof of Proposition <ref> shows that if  is unstable or if the first Chern class of  is even,  ×ℓ^2: [M]_i→ [M]_i+2 fails to have maximum rank in some degree if and only if ℓ is a jumping line. The same proof shows that ×ℓ_1ℓ_2: [M]_i→ [M]_i+2 fails to have maximum rank in some degree if and only if at least one of ℓ_1 and ℓ_2 is a jumping line. The same is not true for the case when  is semistable with the first Chern class odd. 

 






§ JUMPING CONICS AND NON-LEFSCHETZ CONICS



In this section we want to classify the elements of the non-Lefschetz locus of conics of M, the first cohomology module of a rank 2 vector bundle over ^2. In the case of lines, the non-Lefschetz elements are exactly the jumping lines by <cit.>. Hence, the first question that we want to address is whether the elements of the non-Lefschetz locus of conics are exactly the jumping conics. Unfortunately, this is not always the case. We will see that every non-Lefschetz conic is a jumping splitting, but the converse is not always true. 

We first recall the notion of a jumping conic for a semistable vector bundle  of rank 2 over ^2, introduced by Vitter in <cit.>.
We can assume that  is normalized. If C is a smooth conic, then  splits over C as

    _|C≅_^1(a)⊕_^1(-a)      if   c_1()=0; 
    _^1(a)⊕_^1(-a-2)     if   c_1()=-1


Vitter in <cit.> generalizes the Grauert-Mulich theorem, and proves the following:
(<cit.>)
	Let  be a semistable normalized rank 2 vector bundle on ^2 and let C be a general smooth conic. Then
	
    _|C≅_^1⊕_^1    if   c_1()=0; 
    _^1(-1)⊕_^1(-1)     if   c_1()=-1.


Hence a smooth jumping conic for a semistable vector  bundle is defined as a smooth conic for which the splitting type differs from the one described in the previous result.
[<cit.>]
	A smooth conic C is a jumping conic for a semistable vector bundle   if 
	
    _|C≅_^1(a)⊕_^1(-a)      if   c_1()=0; 
    _^1(a-1)⊕_^1(-a-1)     if   c_1()=-1

	with a>0.



To extend this definition also for unstable vector bundles, we first we need to study how an unstable vector bundle  splits when restricted to a general smooth conic.

  
 
 	Let  be an unstable normalized rank 2 vector bundle on ^2, with instability index k, and let C be a general smooth conic. Then
 	
    _|C≅_^1(2k)⊕_^1(2k)      if   c_1()=0; 
    _^1(2k)⊕_^1(-2k-2)     if   c_1()=-1

  
 
 	Let  be an unstable normalized rank 2 vector bundle on ^2, with instability index k, and let C be a general smooth conic.
 	We know that h^0(^2,(-k))≠ 0, and any non-zero section s∈^0(^2,(-k)) is regular, so its vanishing locus has codimension at least 2. Then a
 	 non-zero section s∈^0(^2,(-k)) gives us the exact sequence
 	
    0→_^2→(-k)→ℐ→ 0

 	 where ℐ is the ideal sheaf of a set of points. Then shifting by k we get the sequence
 	
    0→_^2(k)→→𝒬→ 0

 	where we define 𝒬=ℐ(k).
 	Since C is general, we can assume C does not meet the zero locus of s, therefore restricting to C the previous sequence, we have the exact sequence
 	
    0→_^2|C(k)→_|C→𝒬_|C→ 0.
 
 	C is a smooth conic so _^2|C(k)≅_^1(2k) and 𝒬_|C is a line bundle over C≅^1, so there exists an ℓ such that 𝒬_|C≅_^1(ℓ).
 	Then our sequence becomes 
 	
    0→_^1(2k)→_|C→_^1(ℓ)→ 0.

 	We can use this sequence to compute the Euler characteristic of _|C
 	
    χ(_|C)=χ(_^1(2k))+χ(_^1(2k))=2k+l+2.

 	Comparing with
 	
    χ(_|C)=
     		2      if   c_1()=0; 
    
     		0     if   c_1()=-1;

 	we obtain
 	
    ℓ=
     		-2k      if   c_1()=0; 
    
     		-2k-2     if   c_1()=-1.

 	In both cases, the sequence 
 	
    0→_^1(2k)→_|C→_^1(ℓ)→ 0

 	is a split exact sequence since (_^1(2k),_^1(ℓ))=0, so
 	
    _|C≅_^1(2k)⊕_^1(2k)      if   c_1()=0; 
    _^1(2k)⊕_^1(-2k-2)     if   c_1()=-1.

 
 So we can extend Definition <ref> for any rank 2 vector bundle on ^2 as follows:
 
 	A smooth jumping conic for  is a smooth conic C such that the restriction of  to C does not split as to a general conic.
 

 
Until now, we have considered only smooth conics. The next question is how to extend the definition of jumping conic to include also the singular case. When  is semistable, we recall the definition in <cit.>:
[<cit.>]
Let  be a rank 2 normalized semistable bundle. A singular conic C=ℓ_1ℓ_2 is a jumping conic if 

    
  * h^0(;_|C) > 0, when c_1()=-1;
    
  * if either ℓ_1 or ℓ_2 is a jumping line, when c_1()=0.
 


    In the case c_1()=-1, this definition agrees with the one for smooth jumping conics; in fact, we can say that a conic C is a jumping conic if and only if h^0(;_|C) > 0.

Vitter also shows that for singular conics this is equivalent to the following. 
 C=ℓ_1ℓ_2  is a jumping conic for a semistable normalized vector bundle 
exactly when one of the following is true:

	
  * ℓ_1 or ℓ_2 is a jumping line,
	
  * = ℓ^2 and ℓ is a jumping line
	of the second kind (Definition <ref>),
	
  * ℓ_1 and ℓ_2 are generic so that _|ℓ_j=_^1⊕_^1(-1)
	j = 1, 2 and the _^1 summands coincide at the intersection point.
 

We can extend this definition to the case when  is unstable in a natural way:

Let  be a rank 2 normalized unstable vector bundle. A singular conic C=ℓ_1ℓ_2 is a jumping conic if ℓ_1 or ℓ_2 is a jumping line.



To connect the notion of jumping conic with the non-Lefschetz conics let us first state some facts that we will use in the next proof. Recall that a conic  is a  Lefschetz conic for M=_*^1(^2, ) if and only if 
    × C: [M]_i _*^1(^2, (i))→[M]_i+2_*^1(^2, (i+2))
 has maximum rank in for each i. Since all these properties are independent of the shift, we can always assume  normalized.

With a similar argument to in the proof of Theorem <ref>, from the short exact sequence

    0→(i-2)→(i)→(i)_| C→0.

we get the long exact sequence 

    0   →^0(^2,(i-2))→^0(^2,(i))→^0(^2,(i)_| C)→
       →^1(^2,(i-2))×  C→^1(^2,(i))
    	→^1(^2,(i)_| C)→
       →^2(^2,(i-2))→^2(^2,(i))→^2(^2,(i)_| C)=0.

Then we have the following facts:

	
  *  ×  C is injective if h^0(^2,(i)_| C)=0 (sufficient condition but not necessary);
	
  *  ×  C is injective if and only if h^0(^2,(i)_| C)- h^0(^2,(i))+h^0(^2,(i-2))=0;
	
  *  ×  C is injective if and only if the map ^0(^2,(i))→^0(^2,(i)_| C) is surjective;
	
  *  ×  C is surjective if h^1(^2,(i)_| C)=0 (sufficient condition but not necessary);
	
  *  ×  C is surjective if and only if h^1(^2,(i)_| C)- h^2(^2,(i-2))+h^2(^2,(i))=0;
	
  *  ×  C is surjective if and only if the map ^1(^2,(i)_| C)→^2(^2,(i-2)) is injective.

From this point forward we treat separately the cases  unstable, and  stable.


	Let  be an unstable vector bundle. The conic C is not a Lefschetz conic if and only if it is a jumping conic.

 As we saw in Remark <ref>  a singular conic C=ℓ_1ℓ_2 is a non-Lefschetz conic if and only if at least one between ℓ_1 or ℓ_2 is a jumping line, i.e. if and only if C is a jumping conic by definition.
Assume C is a smooth conic and, without loss of generality,  is normalized. 
	
	For  unstable with instability index k we know that 
	
    _| C≅_^1(a)⊕_^1(-a)      if   c_1()=0; 
    _^1(a)⊕_^1(-a-2)     if   c_1()=-1,
 and C is a jumping conic if and only if a≠ 2k. 

	Let us first prove that for any smooth conic C, we must have a≥ 2k.
	From the short exact sequence 
	
    0→(-k-2)→(-k)→(-k)_| C→0,

	we  have the cohomology sequence 
	
    0→^0(^2,(-k-2))→^0(^2,(-k))→^0(^2,(-k)_| C)→…

	By definition of instability index we have that ^0(^2,(-k))≠0 and, for any b>k, ^0(^2,(-b))=0. In particular ^0(^2,(-k-2))=0, so the map ^0(^2,(-k))→^0(^2,(-k)_| C) is injective.
	It follows that  ^0(^2,(-k)_|C)≠ 0. Since
	
    ^0(^2,(-k)_| C)=^0(^1,_^1(-2k+a)⊕_^1(-2k-a) )     if   c_1()=0; 
    ^0(^1,_^1(-2k+a)⊕_^1(-2k-a-2))     if   c_1()=-1,

	we can conclude that a≥ 2k. 
	
	Case 1: c_1()=0. 
	In this case k>0 and ^∨=.
	Using <cit.>  we have for i<k
	
    h^0(^2,(i)_|ℓ)- h^0(^2,(i))+h^0(^2,(i-2))
       =h^0(^2,𝒪_ℙ^1(2i-a))+ h^0(^2,𝒪_ℙ^1(2i+a))-k+i+22+k+i2
       =
    			4t+2      if   i≥a/2;
    
    			a+2i+1      if  -a/2≤ i<a/2;
    
    			0      if  i<-a/2.
    		+
    			-2k-2i-1     if  -k≤ i<k;
    
    			0      if  i<-k;
    		
       =
    			a-2k      if   -k≤ i<k;
    
    			a+2i+1      if  -a/2≤ i<-k;
    
    			0      if  i<-a/2.

	
	
	If a=2k then h^0(^2,(i)_|ℓ)- h^0(^2,(i))+h^0(^2,(i-2))=0 for every i<k and so by Fact (<ref>), the map ×  C is injective for i<k.
 
	When a>2k, using Fact (<ref>)  we get instead that  ×  C is not injective for -a/2≤ i<k (and it  is injective for i< -a/2).
	
	In a similar way using Serre Duality, and <cit.>, for i≥-k
	
    h^1(^2,(i)_| C)- h^2(^2,(i-2))+h^2(^2,(i))=
       h^1(^2,𝒪_ℙ^1(2i-a))+ h^1(^2,𝒪_ℙ^1(2i+a))- h^0(^2,(-i-3))+h^0(^2,(-i-1))=
       h^0(^2,𝒪_ℙ^1(-2i+a-2))+ h^0(^2,𝒪_ℙ^1(-2i-a-2))-k-i+1 2-k-i-12
       = 
    			0    if   i>a/2-1;
    
    			-2i+a-1      if  -a/2-1< i≤a/2-1;
    
    			-4t-2      if  i≤-a/2-1.
    		+
    			0     if  i≥ k;
    
    			2i-2k+1     if -k≤ i<k .

		If a=2k,	using Fact (<ref>),   ×  C is surjective for   i≥-k, so in this case  ×  C  always has maximum rank.
	If a>2k, × C is  not surjective for -k≤ i< a/2. So  ×  C does not have maximum rank for -k ≤ i<k; since k>0 this interval is not empty. Hence a smooth conic C is not a Lefschetz conic for , when a>2k.
	
This proves that for   unstable, normalized with c_1()=0, C is a Lefschetz conic if and only if C is not a jumping conic.
	
	
	Case 2: c_1()=-1. In this case  is unstable and normalized with c_1()=-1, therefore k≥0 and ^∨=(1). We can proceed in a similar way to Case 1.	
 
 Using <cit.>  we have 
	
    h^0(^2,(i)_| C)- h^0(^2,(i))+h^0(^2,(i-2))
       =h^0(^2,𝒪_ℙ^1(2i+a)⊕𝒪_ℙ^1(2i-a-2))- h^0(^2,(i))+h^0(^2,(i-2))
       =
    			4t      if   i> a/2;
    
    			2i+a+1      if  -a/2≤ i≤a/2;
    
    			0      if  i<-a/2.
    		+
    		
    			-2k-2i-1     if  -k≤ i≤ k;
    
    			0      if  i<-k;
    		
       =
    			a-2k      if   -k≤ i≤ k;
    
    			2i+a+1      if  -a/2≤ i<-k;
    
    			0      if  i<-a/2.

	Using Serre Duality and <cit.> again, we get, for i≥ -k
	
	
	
    h^1(^2,(i)_| C)- h^2(^2,(i-2))+h^2(^2,(i))=
    
    		   h^0(^2,𝒪_ℙ^1(-2i-a-2))+h^0(^2,𝒪_ℙ^1(-2i+a))
      -h^0(^2,(-i))+h^0(^2,(-i-2))
       = 
    			0    if   i>a/2;
    
    			-2i+a+1      if  -a/2≤ i≤a/2;
    
    			-4t      if  i<-a/2.
    		+
    			0     if  i≥ k;
    
    			-2i-2k-1     if -k≤ i< k .
    		
       =
    			0    if   i>a/2;
    
    			2i-a-1      if   k<i≤a/2;
    
    			a-2k      if  -k≤ i≤ k.

We can conclude, using Fact (<ref>) and Fact (<ref>) that 
  
      
  * if a=2k,  ×  C is always injective for i≤ k and  surjective for any i≥ -k.  So ×  C always has maximum rank.
      
  * ×  C does not have maximum rank for -k≤ i≤ k, and since k≥ 0, this interval is not empty. So in this case C is not a Weak Lefschetz element.
  
This concludes our proof: C is a Lefschetz conic if only if it is not a jumping conic.



Let us consider now the case  semistable.
When  is semistable with first Chern class c_1()=-1, as noted by Vitter <cit.>, the jumping conics (smooth or singular) are exactly the ones for which h^0(;_|C) ≠0, and equivalently h^1(;_|C) ≠0, while this is not true if c_1()=0. This difference is noticed in the proof <cit.>. 
This is exactly where the difference between jumping conics and non-Lefschetz conic lies: while for c_1()=-1 they are equivalent, for c_1()=0 the jumping conics are a subset, corresponding to the condition of  h^1(;_|C) ≠0. 

 <cit.> 
	The set of jumping conics J_2 of a semistable rank 2 vector bundle  on ^2 can be given the scheme structure of a hypersurface in ^5 of degree c_2() if
	c_1()=0 and of degree c_2()-1 if c_1()=-1. Furthermore, the singular jumping
	conics are in the scheme-theoretic closure of the smooth jumping conics.	

The following results can be seen as corollaries of this theorem. 


	Let  be a stable, normalized vector bundle with c_1()=-1. A conic C fails to be a Weak Lefschetz conic if and only if it is a jumping conic.


Let us first consider the case when C=ℓ_1ℓ_2 is singular. If =ℓ^2 we prove in <ref> that C is a non-Lefschetz conic if and only if it is a jumping line of the second type, i.e. h^0(;_|C) ≠0. The same proof shows that C=ℓ_1ℓ_2 is a non-Lefschetz conic if and only if h^0(;_|C) ≠0.

Let us assume now that C is smooth.
	_| C≅_^1(a-1)⊕_^1(-a-1) and C is a jumping conic if and only if a>0.
 In this case χ(_| C)=0, so h^0(^2,_| C)=h^1(^2,_| C), in particular for a jumping conic they are both non-zero. 
 
	Let us assume C is a jumping conic. ^0(^2,)= 0 because  is stable. Then the map 0= ^0(^2,)→^0(^2,_| C)≠0 is not surjective, so by Fact (<ref>) the map ×  C is not injective.
	
	The map ^1(^2,_| C)→^2(^2,(-2)) can not be injective because  ^1(^2,_| C)≠ 0 and ^2(^2,(-2))≅^0(^2,(1))≅^0(^2,(-2))=0, using Serre Duality and stability of . Then ×  C is not surjective by Fact (<ref>). 
 In conclusion, C is not a Lefschetz conic because the multiplication map × C fails to have maximal rank from degree -2 to degree 0.
 
	Let us consider now a smooth non jumping conic C so _| C≅_^1(-1)⊕_^1(-1). 
 Using Fact (<ref>), the map ×  C is injective if h^0(^2,(i)_| C)=h^0(^1,_^1(2i-1)⊕_^1(2i-1))=0, then it is injective for every i≤0.
	×  C is surjective if h^1(^2,(i)_| C)=0 by Fact (<ref>).
	Using again Serre Duality 
	
    h^1(^2,(i)_| C)=   h^1(^1,_^1(2i-1)⊕_^1(2i-1))
       h^0(^1,_^1(-2i-1)⊕_^1(-2i-1))
 
	then it is zero for i>0.
	Then  ×  C always has maximal rank, so C is a Lefschetz element.




	Let  be a semistable, normalized vector bundle with c_1()=0. A smooth conic C fails to be a Weak Lefschetz conic if and only if it is a jumping conic and _| C≅_^1(a)⊕_^1(-a) with a>1. When C is singular, the definitions of jumping and non-Lefschetz are equivalent.


The case when C is singular is the same as in Proposition <ref> for unstable vector bundles. In fact C=ℓ_1ℓ_2 is a non-Lefschetz conic if and only if at least one between ℓ_1 or ℓ_2 is a jumping line by Remark <ref> as in the definition of singular jumping conic.

Let us assume first that _| C≅_^1(a)⊕_^1(-a) with a=0 or a=1. We want to show that in this case, C is a Lefschetz conic. 
	Since h^0(^2,(i)_| C)=h^0(^1,_^1(2i+1)⊕_^1(2i-1))=0,  for every i<0,  then Fact (<ref>) implies ×  C injective for every i<0.
	
	The map ×  C is surjective if h^1(^2,(i)_| C)=0 by Fact <ref>.
	Using Serre Duality 
	
    h^1(^2,(i)_| C)=   h^1(^1,_^1(2i+a)⊕_^1(2i-a))
    
    		=   h^0(^1,_^1(-2i-2-a)⊕_^1(-2i-2+a))
 
	then it is zero for i≥0 (here a=0 or a=1).
	Then  ×  C always has maximal rank, so C is a Lefschetz element.
	
	Now we will show that for any smooth conic C such that _| C≅_^1(a)⊕_^1(-a) with a>1, ×  C: ^1(^2, (-2))→^1(^2, ) does not have maximal rank and so C is not a Lefschetz conic.
    Note that if a>1, ^0(^2_| C)≅^0(^2,	_^1(-a)⊕_^1(a))≠ 0 and, using Serre Duality
    
    ^1(^2,_| C)=^1(^1,_^1(-a)⊕_^1(a))=^0(^1,_^1(a-2)⊕_^1(-a-2))≠ 0.

 
	We consider first the case  stable; later we return to the case when  is semistable but not stable. 
	
	By stability ^0(^2,)=0, then ^0(^2,)→^0(^2,_| C)≠0 can not be surjective and so by Fact (<ref>), the map ×  C is not injective. 
 
	Using Serre Duality, and (semi)stability  we also have 
    ^2(^2,(-2))=^0(^2(-1))=0.

	Then the map 0≠^1(^2,_| C)→^2(^2,(i-2))=0 is not injective. Then ×  C can not be surjective, by Fact (<ref>). 
 

    
 	When  is semistable but not stable the same argument shows that ×  C is not surjective, but in this case ^0(^2,)≠0. By Fact (<ref>), the map ×  C  is injective if and only if the map ^0(^2,)→^0(^2,_| C) is surjective. Looking at the long exact sequence 
	
    ^0(^2,(-2))→^0(^2,)→^0(^2,_| C)→^1(^2,(-2))×  C→^1(^2,)
    		→…

	we know that that map is injective because ^0(^2,(-2))=0. So ×  C  is injective if and only if the map ^0(^2,)→^0(^2,_| C) is a isomorphism, but this is not possible since h^0((^2,_| C))=a+1>2 while we will show that h^0(^2,)=1.

 We know that h^0(^2,)≠ 0, so we can take  a non-zero section s∈^0(^2,).
 	The section s must be regular, so we have the exact sequence
 	
    0→_^2→→ℐ→ 0

 	where ℐ that is the ideal sheaf of a set of point Z. We assumed M=^1_*(^2,)≠0, hence ≇_^2⊕_^2. This implies that Z≠∅, then  
  h^0(^2,ℐ)=0.
  From the cohomology sequence
  
    0→^0(^2,_^2)→^0(^2,)→^0(^2,ℐ)= 0

  we obtain h^0(^2,)=h^0(^2,_^2)=1.






 §.§ Expected codimension of the non-Lefschetz locus of conics

In this section, we want to talk about the codimension of the non-Lefschetz locus of conics _M.  The non-Lefschetz locus of conics _M is defined a priori as a union of determinantal schemes. To compute the expected codimension, we need to show _A is “concentrated” in one degree.


    The non-Lefschetz locus of M=^1_*(^2,), for any rank 2 vector bundle , as a subscheme of ^5 coincides with the non-Lefschetz locus in the middle degree:
	
    _M =_⌊d-3/2⌋-1,M

 where -d=c_1() is the first Chern class of .


    The proof is similar to what we did for Gorenstein algebras in Theorem <ref>.
    We know that M has the WLP by <cit.>; as a consequence we have that:
    
        
  * the Hilbert function of M is unimodal;
        
  * M has no socle until degree ⌊d-3/2⌋ and no new generators after that degree.
    
    Moreover by Serre Duality and the fact that =(d) we have 
    ^1(^2,(⌊d-4/2⌋-i))≅^1(^2,(-⌊d-3/2⌋+i-3))≅^1(^2,(⌊d-2/2⌋+i)).

    Then we can apply Proposition <ref> and Remark <ref> and follow the same proof then in Theorem <ref> to obtain that 
    _M =_⌊d-3/2⌋-1,M.


This result assures us that the non-Lefschetz locus of conics has a determinantal structure, so we can compute the expected codimension.

 Let  a rank 2 vector bundle over ^2. Then the non-Lefschetz locus of conics of M=^1_*(^2,) has expected codimension 
    
    _M= 
        1    if  is unstable or has first Chen class even;
    
    	2    if  semistable but not stable;
    
    	3    if  stable with c_1() odd.
	


Since _M =_⌊d-3/2⌋-1,M, we have that

    _M= h_⌊d-3/2⌋+1-h_⌊d-3/2⌋-1+1

where -d=c_1() is the first Chern class of .

Case 1: odd Chern class If d is odd, by Serre duality,  h^1 (^2, ( d-3/2 -1 ) )= h^1 (^2, ( d-3/2 +1 ) ), hence _M=1.

Case 2: even Chern class. Let us consider now d even. Without loss of generality, we can assume . Then d=-c_1()=0, and 
_M= h_-1-h_-3+1

Case 2.1:  unstable. If  is unstable with index of instability k, then  by <cit.> the multiplication map for a general line ×ℓ: [M]_i-1→ [M]_i, is  bijective for -(k+1) ≤ i ≤ k-1. This implies 
 h_-k-2=h_-k-1=… =h_k-1, and in particular h_-3=h_-1 since k>0
when  unstable with c_1()=0. So also in this case _M=1.

Case 2.1:  semistable. Let us now consider  semistable with c_1()=0. Then applying Serre Duality we get

    (_M ) =    h_-1-h_-3+1 = h^1(^2,(-1))-h^1(^2,(-3))+1
    
     =    h^1(^2,(-2))-h^1(^2,)+1
    
    	=    -χ((-2))+h^0 (^2,(-2))+h^2 (^2,(-2))
       +χ ( )-h^0 (^2,)-h^2 (^2, )+1
    
    	=    χ ()-χ ((-2))+h^0 (^2, )

since by semicontinuity h^0 (^2,(-2))=0 as well as

    h^2 (^2,(-2))=h^0 (^2,(-1))=0
    
      h^2 (^2, )=h^0 (^2,(-3))=0.

We can see that

    χ((i))-χ((i-2))=
    		4t+2      if   c_1()=0; 
    
    		4t     if   c_1()=-1;

 so χ ()-χ ((-2))=2.
If  is stable h^0 (^2, )=0, while if  is semistable (and M≠0) we show in the proof of Corollary <ref> that h^0 (^2, )=1. Finally, we have

    (_M ) =    χ ()-χ ((-2))+h^0 (^2, )= 2 +h^0 (^2, )
    
     =    
    	2    if  semistable but not stable;
    
    	3    if  stable.



In the previous section, we saw that a general conic is a Lefschetz-conic, hence _M≠^5  and 
so the 1≤ (_M)≤ (_M). This implies that if  is unstable or the first Chen class c_1()=-d is odd, the non-Lefschetz locus of conics of M=^1_*(^2,) is always a hypersurface. In this case the degree is given by h_⌊d-3/2⌋+1.
Note that by Theorem <ref> and Corollary <ref> we already knew that for a semistable bundle  with fist Chern class odd, the non-Lefschetz locus of conics is a hypersurface in ^5 of degree c_2()-1.
  
  
It is left to study the non-Lefschetz locus of conics when  is semistable and c_1()=0. By Theorem <ref>  the set of jumping conics forms a hypersurface in ^5 of degree c_2(), and by Corollary <ref> _M  is contained in this surface.


For a general  semistable vector bundle with the first Chern class even, the non-Lefschetz locus has expected codimension  
    _M= 
    	2    if  semistable but not stable;
    
    	3    if  stable.
	

We will prove this conjecture in the case when  is the syzygy bundle of a complete intersection. We will also show that the hypothesis of generality is necessary, using examples of monomial complete intersections.




§ GENERAL COMPLETE INTERSECTIONS OF HEIGHT 3


In this section, we focus on Artinian complete intersections.
Let A=k[x_1,x_2,x_3]/(f_1,f_2,f_3) be a complete intersection of type (d_1,d_2,d_3), and let  be its first syzygy bundle. Our goal is to prove Conjecture <ref> for . Recall that A≅_*^1(^2, ). Moreover,  has first Chern class odd if and only if the socle degree e is even, and

    
  *   is stable if d_3< d_1+d_2
    
  *  is semistable if d_3≤ d_1+d_2
    
  *  is unstable if d_3> d_1+d_2.

 

	Let A=R/(f_1,f_2,f_3) be a general complete intersection of type (d_1,d_2,d_3). Then, the non-Lefschetz locus of conics has the expected codimension in ^5:
	
    _A= 1    if  e  is even;
    
    		1    if  d_3> d_1+d_2;
    
    		2    if  d_3=d_1+d_2;
    
    		3    if  e  odd d_3≤ d_1+d_2-2.



Note that when either the vector bundle  is unstable or its first Chern class is odd, we know that the expected codimension is achieved and _A is a hypersurface. This happens when either the socle degree e is even or d_3> d_1+d_2. Therefore, in this case, the non-Lefschetz locus of conics is a hypersurface of degree h_⌊e/2⌋-1.

By this reasoning, we can restrict to the case when the socle degree e is odd and d_3≤ d_1+d_2. Equivalently, the first syzygy bundle  is semistable with the first Chern class even.

Let (1,3,h_2,…,h_e-2,3,1) be the Hilbert function of a complete intersection of type (d_1,d_2,d_3).

We will construct a Gorenstein algebra R/J with the same Hilbert function 
    (1,3,h_2,…,h_e-2,3,1)
 such that the non-Lefschetz locus of conics has expected codimension.
We know that when the Hilbert function is fixed, the height 3 Gorenstein algebras with such Hilbert function lie in a flat family <cit.>. Then, by semicontinuity, we can conclude that the general Gorenstein algebra with that Hilbert function has non-Lefschetz locus of expected codimension.
Moreover, from <cit.> we know the Gorenstein algebras with minimum number of generators and Hilbert function (1,3,h_2,…,h_e-2,3,1) form a Zariski dense set in the family of Gorenstein algebras with this Hilbert function. 
In our case, we know that  (1,3,h_2,…,h_e-2,3,1) is the Hilbert function of a complete intersection, so in this family of Gorenstein algebras the ones with minimum number of generators must be the complete intersections. Since this set is dense, we can conclude by semicontinuity that the general complete intersection has the non-Lefschetz locus of conics of expected codimension, assuming that we have constructed a Gorenstein Algebra with the same Hilbert function and non-Lefschetz locus of conics of expected codimension.



 §.§ Proof of Theorem toc<ref>main CItoc<ref>


Before proceeding with the proof of Theorem <ref>, we need to recall some definitions and results about Gorenstein algebras of height 3.

Recall that a sequence (1,3,h_2,…,h_e) is a Hilbert function for a height 3 Gorenstein algebra if and only if it is a  SI-sequence.

	A height 3   SI-sequence is a sequence  (1,3,h_2,…,h_e) such that 
		
  * it is symmetric;
		
  * the first difference (1,2,h_2-h_1,…,h_⌊e/2⌋-h_⌊e/2⌋-1) satisfies Macaulay's growth condition.
	

The second condition is equivalent to the statement that  
    (1,3,h_2,…,h_⌊e/2⌋, h_⌊e/2⌋, …, h_⌊e/2⌋, … )
 is the Hilbert function of zero-dimensional scheme Z on ^2.

For a fixed SI-sequence, the family of Gorenstein algebras R/I having that sequence as a Hilbert function is an irreducible family by <cit.>.


    Let (1,3,h_2,…,h_e) be the Hilbert function of a Gorenstein algebra A. The g-vector of A is the positive part of the first difference: 
    (1,2,g_2,…,g_⌊e/2⌋)=(1,2,h_2-h_1,…,h_⌊e/2⌋-h_⌊e/2⌋-1)



	We say that the g-vector (1,2,g_2,…,g_⌊e/2⌋) is of decreasing type if it begins with (1,2,3,…) (growing as the polynomial ring k[x_1,x_2]) then possibly flat, then strictly decreasing.



	We say that the Gorenstein algebra A with Hilbert function (1,3,h_2,…,h_e), comes from points if it is a quotient of R/I_Z  where I_Z is the ideal associated to a  reduced zero-dimensional scheme Z with Hilbert function 
    (1,3,h_2,…,h_⌊e/2⌋,h_⌊e/2⌋…).



For any SI-sequence, there is always a subfamily of Gorenstein algebras with that sequence as a Hilbert function which comes from points by <cit.>.

	A general complete intersection R/(f_1,f_2,f_3) comes from points if and only if d_3≥ d_1+d_2-1. In this case, [R/(f_1,f_2,f_3)]_j= [R/(f_1,f_2)]_j for j<e-1/2 and the Hilbert function of R/(f_1,f_2) stabilizes at d_1+d_2-2≤e-1/2. 
 

The last definition that we need to recall is a strong form of general position for sets of points Z in ^2:

    Let Z be a set of points in ^2, we say that Z has the Uniform Position Property (UPP) if, for any n≤ Z, all subsets of n points have the same Hilbert function. 

As a consequence, if a set of points Z in ^2 has the UPP, and Y is a subset of n points of Z we have that the Hilbert function h_i(Y)=[R/I_Y] of Y must be the truncation of the Hilbert function of Z: 
    h_i(Y)=min{h_i(Z), n},
 where h_i(Z)=[R/I_Z]_i is the Hilbert function of Z

Now we can proceed with the proof of Theorem <ref>: we want to show that the non-Lefschetz locus of conics _A of a general complete intersection A=R/(f_1,f_2,f_3) of type (d_1,d_2,d_3) has expected codimension. The only case left to prove is when the socle degree e is odd and d_3≤ d_1+d_2.

 
Let A=R/(f_1,f_2,f_3) a complete intersection of type (d_1,d_2,d_3), with e is odd and d_3≤ d_1+d_2. Let (1,3,h_2,…,h_e-2,3,1) be the Hilbert function of A.
By Proposition <ref>,  the expected codimension of _A is

    
  * 2 when d_3=d_1+d_2 (this corresponds to the case when the syzygy bundle  is semistable but not stable);
    
  * 3 otherwise (when the syzygy bundle  is semistable with c_1() even).


First, we show that the g-vector is always of decreasing type. Assume by contradiction that

    (1,2,g_2,…,g_⌊e/2⌋)=(1,2,h_2-h_1,…,h_⌊e/2⌋-h_⌊e/2⌋-1)

is not of decreasing type. Then there exists i<e-1/2 such that g_i-2>g_i-1=g_i. Since g_i-2>g_i-1, we have that d_1, d_2 ≤ i. Now g_i-1=g_i so by <cit.> all the generators of degree ≤ i have a common factor of degree g_i. So f_1 and f_2 have a common factor, but this is not possible since (f_1,f_2,f_3) is a complete intersection.


We will now construct a Gorenstein algebra R/J with the same Hilbert function of a complete intersection of type (d_1,d_2,d_3), such that the non-Lefschetz locus of conics has expected codimension. 

Let R/J be an Artinian Gorenstein algebra with  Hilbert function 
    (1,3,h_2,…,h_e-2,3,1),
  that comes from points. So R/J is obtained as a quotient of R/I_Z where Z is a reduced zero-dimensional scheme on ^2.
Therefore [R/I_Z]_i=[R/J]_i for every i≤e-1/2; moreover, since h_e-1/2=h_ e+1/2 and the Hilbert function of I_Z stabilises at e-1/2, we also have that 
[R/I_Z]_e+1/2=[R/J]_e+1/2.

Since we showed that the positive first difference of  (1,3,h_2,…,h_e-2,3,1) is of decreasing type, we can assume Z  satisfies the UPP by <cit.>.

By Theorem <ref>, C_R/J is “concentrated” in the middle degree, i.e. _R/J= _R/J,e-1/2-1. Then under the identification

 
    [R/J]_e-1/2-1r× C[equal]d   [R/J]_e+1/2[equal]d
    [R/I_Z]_e-1/2-1r× C   [R/I_Z]_e+1/2

C is a non-Lefschetz conic for R/J if and only if × C: [R/I_Z]_e-1/2-1→ [R/I_Z]_e+1/2 is not injective.
If C does not pass through any of the points of Z, then C is a non-zero divisor and so the multiplication map is injective. Let us assume that C passes through at least one point of Z and let I_Y=(I_Z:C) be the ideal associated to the set of points Y of Z that C does not pass through. Then the map × C: [R/I_Z]_e-1/2-1→ [R/I_Z]_e+1/2 is injective if and only if [I_Y/I_Z]_e-1/2-1=0.
So we want to check whether the Hilbert function of I_Y and I_Z are equal in degree e-1/2-1.

Since we assumed Z has the UPP, the Hilbert function of  I_Y depends only on the number of points of I_Y, and so it must be the truncated Hilbert function of Z; in particular

    [R/I_Y]_e-1/2-1=min{[R/I_Z]_e-1/2-1, n}.



Case 1: d_3= d_2+d_1. In this case _A =2, hence h_e-1/2-1-h_e+1/2=1.
Since  
    ([R/I_Z]_e-1/2)-([R/I_Z]_e-1/2-1)=h_e-1/2-1-h_e-1/2=h_e-1/2-1-h_e+1/2=1,

if C meets just one of the points of Z then [I_Y/I_Z]_e-1/2-1=0. If C passes through exactly 2 points of Z, then [I_Y]_e-1/2-1= [I_Z]_e-1/2-1-1 and so the map × C is not injective.
We can conclude that C∈_R/J if and only if C passes through at least 2 points of Z.
Then _R/J=2 as we wanted.

Case 2: d_3≤ d_2+d_1-2.   Here _A =3, hence h_e-1/2-1-h_e+1/2=2.

 In this case 
    ([R/I_Z]_e-1/2)-([R/I_Z]_e-1/2-1)=h_e-1/2-1-h_e-1/2=h_e-1/2-1-h_e+1/2=2,
 so
  [I_Y]_e-1/2-1≠ [I_Z]_e-1/2-1 if C passes through at least 3 points of Z. 
 We can conclude that _R/J=3 as expected. 
 
 By semicontinuity, the non-Lefschetz locus of a general Gorenstein ideal with Hilbert function  (1,3,h_2,…,h_e-2,3,1) has codimension 3. In particular, for a general complete intersection I of type (d_1,d_2,d_3), the non-Lefschetz locus has expected codimension.  







 §.§ Note about the case tocd_3≤ d_1+d_2-4tocd_3≤ d_1+d_2-4 and odd socle degree



In Remark <ref> we notice that for d_3≥ d_1+d_2-1, the Artinian complete intersection A=R/(f_1,f_2,f_3) comes from points. Moreover, if Z is the set of points defined as a zero-dimensional scheme by the ideal (f_1,f_2), we have that 
[A]_j=[R/(f_1,f_2)]_j for any j<d_3. Since d_3≥ d_1+d_2-1

    *e/2+1≥d_3+d_3-2/2+1=d_3-1.


By Theorem <ref>, _A= _A,⌊e/2⌋-1, so C is a non-Lefschetz conic if and only if
the map
 
    [R/(f_1,f_2,f_3)]_⌊e/2⌋-1r× C[equal]d   [R/(f_1,f_2,f_3)]_⌊e/2⌋+1[equal]d
    [R/(f_1,f_2)]_⌊e/2⌋-1r× C   [R/(f_1,f_2)]_⌊e/2⌋+1

 is not injective. Then a conic C that is in the non-Lefschetz locus of conics of A=R/(f_1,f_2,f_3) needs necessarily to vanish at least at one of the points of Z (this condition is not necessarily sufficient, as we can see in the case d_3=d_1+d_2). 
 
 This is not true in general. In fact, the subset of the non-Lefschetz locus of conics that do not vanish at any point of Z for a general complete intersection A=R/(f_1,f_2,f_3) with d_3≤ d_1+d_2-4 and e odd, has codimension  3 in ^5. This agrees with the codimension of the entire scheme _A by Theorem <ref>. 


	Let A=R/(f_1,f_2,f_3) be a general complete intersection with d_3≤ d_1+d_2-4 and e odd.
	Assume Z is the set of points defined as a zero-dimensional scheme by the ideal (f_1,f_2).
	The set of conics C  in the non-Lefschetz locus of conics _A that do not vanish at any of the points of Z has codimension 3 in ^5.


	Let A=R/(f_1,f_2,f_3) be a general complete intersection and let Z be the set of points defined as a zero-dimensional scheme by the ideal (f_1,f_2). For this proof, we consider just the set of conics C that do not pass through any of the points of Z, or equivalently, the conics such that (C,f_1,f_2) is a complete intersection. As before (1,3,h_2,…,h_e-2,3,1) is the Hilbert function of R/I. Here, we are also assuming the socle degree e odd.
	
	
	
	To check if a conic C is a Lefschetz conic it is enough to verify whether the multiplication map 
    × C: [A]_e-1/2→ [A]_e+1/2+1
 is surjective, or equivalently if   [R/(C,f_1,f_2,f_3)]_e+1/2+1 is zero.
	A general conic C is a Lefschetz conic, so [R/(C,f_1,f_2,f_3)]_e+1/2+1=0. In this case (C,f_1,f_2,f_3) is an almost complete intersection of type (2,d_1,d_2,d_3) with Hilbert function 
    (1,3,h_2-1,h_3-h_1,…,h_e-1/2-h_e-1/2-2,h_e+1/2-h_e-1/2-1).

	We want to look now at the conics C∈_R/I, so that [R/(C,f_1,f_2,f_3)]_e+1/2+1≠0.
	Since we are just considering conics that don't vanish at any point of Z, the ideal (C,f_1,f_2) is a complete intersection. 
	To compute the codimension it is enough to consider C such that (C,f_1,f_2,f_3) is an almost complete intersection whose Hilbert function differs from the general case in the least possible way. Note that if 
    [R/(C,f_1,f_2,f_3)]_e+1/2+1=1,
 by duality we also have 
    (× C: [R/(C,f_1,f_2,f_3)]_e-1/2-1→[R/(C,f_1,f_2,f_3)]_e+1/2)=1,
  [R/(C,f_1,f_2,f_3)]_e+1/2=h_e+1/2-h_e-1/2-1+1 Then we can assume that the Hilbert function of R/(C,f_1,f_2,f_3) is  
	
    (1,3,h_2-1,h_3-h_1,…,h_e-1/2-h_e-1/2-2,h_e+1/2-h_e-1/2-1+1,1).

	We want to compute the codimension of the almost complete intersection of type (2,d_1,d_2,d_3) in R with Hilbert function  
    (1,3,h_2-1,h_3-h_1,…,h_e+1/2-h_e-1/2-1+1,1)

	in the space of all almost complete intersection of type (2,d_1,d_2,d_3).
	A general almost complete intersection J  of type (2,d_1,d_2,d_3) has Hilbert function 
    (1,3,h_2-1,h_3-h_1,…,h_e+1/2-h_e-1/2-1).

	We can link J by a complete intersection K of
	type (2,d_1, d_2) to a Gorenstein ideal G with socle degree s = d_1 + d_2  - d_3 - 1 and
	h-vector (1, f_1, … , f_s). In a similar way, we link a complete intersection J'  with Hilbert function  (1,3,h_2-1,h_3-h_1,…,h_e+1/2-h_e-1/2-1+1,1) by a complete intersection K of type (2,d_1, d_2) to a Gorenstein ideal G' with socle degree s and h-vector (1, f'_1, …, f'_s ) that differs from the one of G only in the middle degrees. In fact, since the Hilbert function of J and J' differ by one in degree e/2 and e/2+1 and are equal otherwise, we can relate the Hilbert function of G and G' as follows:
	
    f'_i=f_i-1     if  i=s-1/2 or  i=s+1/2;
    
    		f'_i=f_i     otherwise.
.
	 
	
	We follow a similar method to the proof of <cit.>. Using the same notation we denote by
	
		
  * D_1, resp. D_1', the dimension of the family of Gorenstein Artinian ideals G, resp. G', with Hilbert function (1, f_1, …, f_s ), (1, f'_1, …, f'_s ) respectively.
		
  * D_2, resp. D_2', the dimension of complete intersections K of type (2,d_1, d_2) contained in G, G' respectively; 
		
  * D_3, resp. D_3', the dimension of complete intersections K of type (2, d_1, d_2) contained in J, J' respectively.
	
	Then the codimension of the almost complete intersection that we are looking at is exactly   
	D_1+D_2-D_3 -( D'_1+D'_2-D'_3).
	The reason why we subtract D_3 (D'_3 resp.) is to remove over-counting, since the same ideal J (J')
	can be reached from many different ideals G (G') using different complete intersections in J (J').
	
	
	 Since the Hilbert function J and J' are different only in degree e+1/2 and e-1/2+1, D_3≠ D_3' only in the case when one of the degrees 2,d_1, d_2 is equal to e+1/2 or e+1/2+1. But this is not possible because 
	d_3≤ d_1+d_2-4, therefore 2≤ d_1≤ d_2 ≤ d_3 < e+1/2. Then D_3=D_3'. 
	
	Similarly D_2≠ D'_2 only when one of the degrees 2,d_1, d_2 is equal to s±1/2, since the Hilbert functions of G and G' differ only in the middle degrees.
	This can happen if and only if d_3= d_1+d_2-4 or d_3= d_1+d_2-6, in both cases D_2-D'_2=-1
	
	Finally to compute D_1-D'_1,  we can apply <cit.>: G and G' are Gorenstein algebras in codimension 3 with socle degree odd and Hilbert function that differs by one only in the middle degree, then 
	D_1-D'_1=f_s-1/2+1-2f_s-1/2+3 +f_s-1/2+4+1.
	Recall that we obtained the Gorenstein Artinian ideal G from J linking by a complete intersection of K of type (2,f_1,f_2).
	Let (1,3,h̃_2,...,h̃_d_1+d_2-1,3,1) the Hilbert function of K, the socle degree is d_1+d_2-1.
	Using the property of linkage and symmetry we have that 
	
    f_s-1/2+1+j=h̃_d_1+d_2+d_3/2+j=h̃_d_1+d_2-d_3-2/2-j=d_1+d_2-d_3-2/2-j+22-d_1+d_2-d_3-2/2-j2;

	where the last equality uses the fact that d_1+d_2-d_3-2/2≤ d_1 and in our case j=0,2,3.
	After some numerical computation, we obtain that 
	
    D_1-D'_1=f_s-1/2+1-2f_s-1/2+3 +s-1/2+4+1=
    		4     if  d_3=d_1+d_2-4 
        or  d_3=d_1+d_2-6;
    
    		3     if  d_3≤ d_1+d_2-8.

	Then we can conclude that the codimension of the conics C  in the non-Lefschetz locus of conics _A that do not vanish at any of the points of Z in ^5 is
	
    (D_1-D'_1)+(D_2-D'_2)-(D_3-D'_3)=
       
    			4    if  d_3=d_1+d_2-4
        or  d_3=d_1+d_2-6;
    
    			3    if  d_3≤ d_1+d_2-8; 
    		 + 
    	
    		-1    if  d_3=d_1+d_2-4
        or  d_3=d_1+d_2-6;
     0    if  d_3≤ d_1+d_2-8.

	is 3 as required. 





§ EXAMPLES WITH MONOMIAL COMPLETE INTERSECTIONS

In this section, we show that the hypothesis of generality in Theorem <ref> is necessary, by constructing examples of monomial complete intersections where the non-Lefschetz locus of conics does not have the expected codimension.
Since we know that for every Artinian monomial complete intersection A=k[x_1,x_2,x_3]/(x_1^d_1, x_2^d_2, x_3^d_3)  the non-Lefschetz locus of conics is a hypersurface in ^5 if d_3>d_1+d_2+1 or if the socle degree e=d_1+d_2+d_3-3 is odd, here we will focus on the case when the socle degree e is even and d_3≤ d_1+d_2.

First, if d_3= d_1+d_2, in this case _A has expected codimension 2, but this codimension is never achieved:

    The non-Lefschetz locus of conics of an Artinian monomial complete intersection A=k[x_1,x_2,x_3]/(x_1^d_1, x_2^d_2, x_3^d_3) with d_3=d_1+d_2 has codimension 1 in ^5.


   As in Case 2.2 of Theorem <ref>, C is a Lefschetz conic if and only if the multiplication map  × C: [A]_e-1/2-1→ [A]_e+1/2 is injective. 
Since e+1/2=d_1+d_2-1=d_3-1<d_3, it is equivalent to check when the map  × C: [R/(x_1^d_1,x_2^d_2)]_d_1+d_2-3→ [R/(x_1^d_1,x_2^d_2)]_d_1+d_2-1 is injective.  We can see that × C is not injective if and only if a_6=0, where 
    C=a_1x_1^2+ a_2x_1x_2+ a_3x_1x_3+ a_4x_2^2+ a_5x_2x_3 + a_6x_3^2.

Clearly if a_6≠ 0, then C is not a zero-divisor in R/(x_1^d_1,x_2^d_2) and × C must be injective. If a_6=0,  we can check that, if a_5≠ 0

    x_1^d_2-1x_2^d_1-1-a_3/a_5x_1^d_2-2x_2^d_1-1× c↦ 0,

while if a_5=0

    x_1^d_2-2x_2^d_1-1× c↦ 0.

So C=a_1x_1^2+ … + a_6x_3^2∈_A if and only if a_6=0, and _A=1.

This proposition gives us that if d_1+d_2=d_3, then the non-Lefschetz locus of conics is defined as a set by the ideal (a_6), but _A as subscheme of ^5 does not need to be reduced or unmixed, as we can see in the following example.


	Using Macaulay2 we obtain that, the non-Lefschetz locus of conics of the monomial complete intersection  
 
    A=k[x_1,x_2,x_3]/(x_1^2, x_2^2, x_3^4)

 is defined by the ideal 
 
    I(C_A)=(a_6^3,a_5a_6^2,-a_3a_6^2,-2a_3a_5a_6+a_2a_6^2).

 While the expected codimension is 2 in this case, in agreement with the previous proposition,
  _A=1 and √(I(_A))=(a_6).
 This ideal is saturated, but is not radical and not unmixed:  the primary decomposition 
 
    {( a_6),(a_5,a_6^2),(a_6^2,a_3a_6,a_3^2),(a_5^2,a_3^2,a_6^3,a_5a_6^2,a_3a_6^2,2a_3a_5a_6-a_2a_6^2)}

 has components of respective codimension 1, 2, 2 and 3.


Let us consider  now monomial complete intersections 
    k[x_1,x_2,x_3]/(x_1^d_1, x_2^d_2, x_3^d_3)

with d_3> d_1+d_2 and socle degree e even. In this case, the expected codimension is 3, but examples show that all possible codimensions are achieved.  

	For the monomial complete intersection A=R/(x_1^2,x_2^2,x_3^2), the non-Lefschetz locus of conics is given by the ideal generated by the coefficients of the square-free monomial of C:
 
    I(_A)=(a_5, a_3, a_2),

 so it has expected codimension: 𝒞_A=3.
 


 For A=k[x_1,x_2,x_3]/(x_1^3, x_2^3, x_3^4),  we have that _A=2.

    I(C_A)=(a_6,a_3^2a_4-a_2a_3a_5+a_1a_5^2)∩ (a_4,a_5^2,a_2a_5,a_2^2,3a_3^2a_5-2a_2a_3a_6-2a_1a_5a_6)
       ∩ (a_5,a_3,a_2^2+2a_1a_4)∩ (a_1,a_3^2,a_2a_3,a_2^2,3a_3a_5^2-2a_3a_4a_6-2a_2a_5a_6)
       ∩ (3a_3a_5^2-2a_3a_4a_6-2a_2a_5a_6,12a_1a_5^2+a_2^2a_6-10a_1a_4a_6,2a_3a_4a_5-a_2a_5^2-6a_2a_4a_6,3a_3^2a_5-2a_2a_3a_6-2a_1a_5a_6,12a_2a_3a_5-7a_2^2a_6-2a_1a_4a_6,12a_3^2a_4+a_2^2a_6-10a_1a_4a_6,8a_2a_3a_4+a_2^2a_5-2a_1a_4a_5,a_2a_3^2-2a_1a_3a_5+6a_1a_2a_6,a_2^2a_3-2a_1a_3a_4+8a_1a_2a_5,a_5^4+4a_4a_5^2a_6-4a_4^2a_6^2,a_3^4+4a_1a_3^2a_6-4a_1^2a_6^2,a_2^4-68a_1a_2^2a_4+4a_1^2a_4^2)
       ∩(a_5^2a_6+2a_4a_6^2,a_4a_5a_6,a_6^4,a_5a_6^3,a_4a_6^3,3a_3a_5a_6^2-2a_2a_6^3,2a_3a_4a_6^2+a_2a_5a_6^2,a_4^3a_6,a_3^2a_4a_6-a_2a_3a_5a_6+a_2^2a_6^2-6a_1a_4a_6^2,a_3^2a_4a_5-a_2a_3a_5^2+a_1a_5^3+a_2^2a_5a_6,a_3^2a_4^2-a_2a_3a_4a_5+a_1a_4a_5^2+a_2^2a_4a_6-4a_1a_4^2a_6,a_5^5,a_4a_5^4,a_4^2a_5^3,a_3a_4a_5^3-a_2a_5^4,a_4^3a_5^2,a_3a_4^2a_5^2-a_2a_4a_5^3,a_2a_3a_4a_5^2-a_2^2a_5^3+a_1a_4a_5^3+2a_2a_3a_4^2a_6,a_4^4a_5,a_3a_4^3a_5-a_2a_4^2a_5^2,a_2a_3a_4^2a_5-a_2^2a_4a_5^2+a_1a_4^2a_5^2,a_4^5,a_3a_4^4-a_2a_4^3a_5,a_3^4a_4-a_2a_3^3a_5+a_1a_3^2a_5^2+a_2^2a_3^2a_6-4a_1a_2a_3a_5a_6+4a_1a_2^2a_6^2-6a_1^2a_4a_6^2,a_2^2a_3^2a_4-a_2^3a_3a_5+a_1a_2^2a_5^2+a_2^4a_6-4a_1a_2^2a_4a_6-18a_1^2a_4^2a_6)
       ∩(a_6^4,a_5a_6^3,a_3a_6^3,a_5^3a_6^2,a_3a_5^2a_6^2,a_3^2a_5a_6^2,a_3^2a_4a_6^2-a_1a_5^2a_6^2,a_3^3a_6^2,3a_3a_5^3a_6-2a_3a_4a_5a_6^2-2a_2a_5^2a_6^2,3a_3^2a_5^2a_6-a_2^2a_6^3-2a_1a_4a_6^3,2a_2a_3a_5^2a_6-a_1a_5^3a_6+2a_2a_3a_4a_6^2-a_2^2a_5a_6^2,3a_3^3a_5a_6-2a_2a_3^2a_6^2-2a_1a_3a_5a_6^2,a_3^3a_4a_6-2a_2a_3^2a_5a_6+a_2^2a_3a_6^2-2a_1a_2a_5a_6^2,14a_2a_3a_4a_5a_6^2+2a_2^2a_5^2a_6^2-6a_1a_4a_5^2a_6^2+a_2^2a_4a_6^3+2a_1a_4^2a_6^3,2a_2^2a_3^2a_6^2+14a_1a_2a_3a_5a_6^2-6a_1^2a_5^2a_6^2+a_1a_2^2a_6^3+2a_1^2a_4a_6^3,a_5^5a_6,a_3^5a_6,2a_3^2a_4a_5^3-2a_2a_3a_5^4+2a_1a_5^5+2a_2^2a_5^3a_6-5a_1a_4a_5^3a_6+6a_2a_3a_4^2a_6^2+3a_2^2a_4a_5a_6^2,a_3^3a_4a_5^2-a_2a_3^2a_5^3+a_1a_3a_5^4-2a_2a_3^2a_4a_5a_6+a_1a_3a_4a_5^2a_6-2a_2^2a_3a_4a_6^2-2a_1a_3a_4^2a_6^2+2a_1a_2a_4a_5a_6^2,a_3^4a_4a_5-a_2a_3^3a_5^2+a_1a_3^2a_5^3+a_1a_3^2a_4a_5a_6-a_1^2a_5^3a_6+4a_1a_2a_3a_4a_6^2-3a_1a_2^2a_5a_6^2-2a_1^2a_4a_5a_6^2,a_3^4a_4^2-2a_2a_3^3a_4a_5+a_2^2a_3^2a_5^2+2a_1a_3^2a_4a_5^2-2a_1a_2a_3a_5^3+a_1^2a_5^4+2a_2^2a_3^2a_4a_6+a_1a_3^2a_4^2a_6-2a_2^3a_3a_5a_6-10a_1a_2a_3a_4a_5a_6+2a_1a_2^2a_5^2a_6+a_1^2a_4a_5^2a_6+a_2^4a_6^2+a_1a_2^2a_4a_6^2-2a_1^2a_4^2a_6^2,a_3^5a_4-a_2a_3^4a_5+a_1a_3^3a_5^2+a_2^2a_3^3a_6-5a_1a_2a_3^2a_5a_6+4a_1a_2^2a_3a_6^2-2a_1^2a_2a_5a_6^2,a_3^3a_5^5,a_5^9,a_3a_5^8,a_3^2a_5^7,a_3^5a_5^4,a_3^6a_5^3,a_3^7a_5^2,a_3^8a_5,a_3^9)
       ∩ (a_3^2a_6+2a_1a_6^2,a_1a_3a_6,9a_1a_3^2+2a_1^2a_6,a_6^4,a_3a_6^3,a_1a_6^3,3a_3a_5a_6^2-2a_2a_6^3,a_2a_3a_6^2+2a_1a_5a_6^2,a_2a_3a_5a_6-a_1a_5^2a_6-a_2^2a_6^2+6a_1a_4a_6^2,a_1^3a_6,9a_3^3a_5+16a_1a_2a_6^2,9a_1a_2a_3a_5-9a_1^2a_5^2-9a_1a_2^2a_6+38a_1^2a_4a_6,9a_1^2a_3a_5+2a_1^2a_2a_6,a_3^3a_4-a_2a_3^2a_5+a_1a_3a_5^2+a_2^2a_3a_6,a_3^4,9a_2a_3^3-16a_1^2a_5a_6,11a_1^2a_2a_3-2a_1^3a_5,a_1^3a_3,a_1^4,a_3^2a_4a_5^2-a_2a_3a_5^3+a_1a_5^4+a_2^2a_5^2a_6-4a_1a_4a_5^2a_6+18a_1a_4^2a_6^2,73a_2a_3^2a_4a_5+8a_2^2a_3a_5^2-81a_1a_3a_4a_5^2-8a_1a_2a_5^3-81a_2^2a_3a_4a_6-8a_2^3a_5a_6+32a_1a_2a_4a_5a_6,a_2^2a_3^2a_4-a_2^3a_3a_5+a_1a_2^2a_5^2+a_2^4a_6-4a_1a_2^2a_4a_6-18a_1^2a_4^2a_6)
       ∩(3a_3a_5^2a_6^3-2a_3a_4a_6^4-2a_2a_5a_6^4,3a_3^2a_5a_6^3-2a_2a_3a_6^4-2a_1a_5a_6^4,2a_2a_3a_5a_6^3+2a_1a_5^2a_6^3-a_2^2a_6^4-2a_1a_4a_6^4,a_3^2a_4a_6^3-a_1a_5^2a_6^3,3a_3a_5^3a_6^2-2a_3a_4a_5a_6^3-2a_2a_5^2a_6^3,3a_1a_5^3a_6^2+6a_2a_3a_4a_6^3+a_2^2a_5a_6^3-4a_1a_4a_5a_6^3,3a_3a_4a_5^2a_6^2-2a_3a_4^2a_6^3-2a_2a_4a_5a_6^3,a_3^2a_5^2a_6^2,3a_3^2a_4a_5a_6^2-3a_2a_3a_5^2a_6^2+2a_2^2a_5a_6^3-2a_1a_4a_5a_6^3,3a_3^3a_5a_6^2-2a_2a_3^2a_6^3-2a_1a_3a_5a_6^3,3a_2a_3^2a_5a_6^2-3a_1a_3a_5^2a_6^2-2a_2^2a_3a_6^3+2a_1a_3a_4a_6^3,3a_1a_3^2a_5a_6^2-2a_1a_2a_3a_6^3-2a_1^2a_5a_6^3,3a_3^3a_4a_6^2+a_2^2a_3a_6^3-4a_1a_3a_4a_6^3+6a_1a_2a_5a_6^3,3a_3^2a_5^3a_6-6a_2a_3a_5^2a_6^2-4a_2a_3a_4a_6^3+2a_2^2a_5a_6^3,3a_2a_3a_5^3a_6-3a_1a_5^4a_6+3a_3^2a_4^2a_6^2-6a_2a_3a_4a_5a_6^2-3a_2^2a_5^2a_6^2+3a_1a_4a_5^2a_6^2-a_2^2a_4a_6^3-2a_1a_4^2a_6^3,6a_1a_3a_5^3a_6-3a_2^2a_3a_5a_6^2-6a_1a_3a_4a_5a_6^2+2a_2^3a_6^3+4a_1a_2a_4a_6^3,3a_3^2a_4a_5^2a_6-3a_3^2a_4^2a_6^2+3a_1a_4a_5^2a_6^2-a_2^2a_4a_6^3-2a_1a_4^2a_6^3,3a_3^3a_5^2a_6-6a_1a_3a_5^2a_6^2-2a_2^2a_3a_6^3+4a_1a_3a_4a_6^3-4a_1a_2a_5a_6^3,6a_2a_3^2a_5^2a_6+6a_2a_3^2a_4a_6^2-9a_2^2a_3a_5a_6^2-6a_1a_3a_4a_5a_6^2+6a_1a_2a_5^2a_6^2+4a_2^3a_6^3+8a_1a_2a_4a_6^3,3a_1a_3^2a_5^2a_6+3a_1a_3^2a_4a_6^2-3a_1^2a_5^2a_6^2-a_1a_2^2a_6^3-2a_1^2a_4a_6^3,
       2a_1a_2a_3a_5^2a_6-a_1^2a_5^3a_6+2a_1a_2a_3a_4a_6^2-a_1a_2^2a_5a_6^2,6a_3^3a_4a_5a_6-3a_2^2a_3a_5a_6^2-6a_1a_3a_4a_5a_6^2+2a_2^3a_6^3+4a_1a_2a_4a_6^3,
       a_3^3a_4^2a_6-2a_2a_3^2a_4a_5a_6+a_2^2a_3a_4a_6^2-2a_1a_2a_4a_5a_6^2,3a_3^4a_4a_6-3a_2a_3^3a_5a_6+3a_2^2a_3^2a_6^2-3a_1a_3^2a_4a_6^2+6a_1a_2a_3a_5a_6^2-3a_1^2a_5^2a_6^2+a_1a_2^2a_6^3+2a_1^2a_4a_6^3,
       a_3^2a_4a_5^3-a_2a_3a_5^4+a_1a_5^5-3a_2a_3a_4a_5^2a_6+a_2^2a_5^3a_6-a_1a_4a_5^3a_6+3a_2^2a_4a_5a_6^2,
       a_3^3a_4a_5^2-a_2a_3^2a_5^3+a_1a_3a_5^4-2a_2a_3^2a_4a_5a_6+2a_2^2a_3a_5^2a_6+a_1a_3a_4a_5^2a_6-a_1a_2a_5^3a_6-2a_1a_3a_4^2a_6^2-a_2^3a_5a_6^2+2a_1a_2a_4a_5a_6^2,
       a_3^4a_4a_5-a_2a_3^3a_5^2+a_1a_3^2a_5^3-a_2a_3^3a_4a_6+2a_2^2a_3^2a_5a_6+a_1a_3^2a_4a_5a_6-a_1^2a_5^3a_6-a_2^3a_3a_6^2+4a_1a_2a_3a_4a_6^2-a_1a_2^2a_5a_6^2-2a_1^2a_4a_5a_6^2,
       a_3^4a_4^2-2a_2a_3^3a_4a_5+a_2^2a_3^2a_5^2+2a_1a_3^2a_4a_5^2-2a_1a_2a_3a_5^3+a_1^2a_5^4+2a_2^2a_3^2a_4a_6+a_1a_3^2a_4^2a_6-2a_2^3a_3a_5a_6-10a_1a_2a_3a_4a_5a_6+2a_1a_2^2a_5^2a_6+a_1^2a_4a_5^2a_6+a_2^4a_6^2+a_1a_2^2a_4a_6^2-2a_1^2a_4^2a_6^2,
       a_3^5a_4-a_2a_3^4a_5+a_1a_3^3a_5^2+a_2^2a_3^3a_6-a_1a_3^3a_4a_6-3a_1a_2a_3^2a_5a_6+3a_1a_2^2a_3a_6^2,
       a_6^8,a_5^4a_6^4,a_3^4a_6^4,27a_3^3a_5^5-16a_2^3a_5^2a_6^3+52a_1a_2a_4a_5^2a_6^3,a_4a_5^3a_6^5-a_4^2a_5a_6^6,
       a_1a_3^3a_6^5-a_1^2a_3a_6^6,a_1^2a_4^3a_6^4,a_2^2a_5^9,a_1^4a_2^2a_4^5,a_2^2a_3^9,a_5^13,a_1^6a_4^7,a_3^13,a_1^13,a_4^16,a_2^17,a_1^7a_2^11,a_2^9a_4^10)
It is interesting to notice that I(_A) is neither saturated nor unmixed: we can show using Macaulay2 that the primary components have codimension respectively 2,3,3,3,3,3,3,3, and 6. The last component is Artinian, and does not correspond to any geometric component. Unlike Example <ref>, the radical is also not unmixed, with the first component having codimension 2 and the rest having codimension 3:
 

    √(I(_A))=   (a_6, a_3^2a_4-a_2a_3a_5+a_1a_5^2)∩(a_5, a_4, a_2)∩ ( a_5, a_3, a_2^2+2 a_1a_4)∩(a_3, a_2, a_1)∩
       (3 a_3a_5^2-2 a_3a_4a_6-2 a_2a_5a_6, 12 a_1a_5^2+a_2^2a_6-10 a_1a_4a_6, 2 a_3a_4a_5-a_2a_5^2-6 a_2a_4a_6,
       3 a_3^2a_5-2 a_2a_3a_6-2 a_1a_5a_6,
        12 a_2a_3a_5-7 a_2^2a_6-2 a_1a_4a_6, 12 a_3^2a_4+a_2^2a_6-10 a_1a_4a_6, 8 a_2a_3a_4+a_2^2a_5-2 a_1a_4a_5, a_2a_3^2-2 a_1a_3a_5+6 a_1a_2a_6,
       a_2^2a_3-2 a_1a_3a_4+8 a_1a_2a_5, a_5^4+4 a_4a_5^2a_6-4 a_4^2a_6^2, a_3^4+4 a_1a_3^2a_6-4 a_1^2a_6^2, a_2^4-68 a_1a_2^2a_4+4 a_1^2a_4^2)

 
 
 
 
 Finally, if A=R/(x_1^4,x_2^4,x_3^6), then the non-Lefschetz locus of conics has 𝒞_A=1.
 Also in this case, the ideal is neither saturated nor unmixed.

 Note that even if 𝒞_A=1 and it is contained in the hypersurface parametrizing the the jumping conics, these do not coincide.
 Using Macaulay2 we can show that √(I(_A)) has one primary component of codimension 1, the ideal (a_6), and all other components have codimension 3.





§ GENERAL GORENSTEIN ALGEBRAS

The construction of a Gorenstein algebra with the non-Lefschetz locus of expected codimension in the proof of Theorem <ref> suggests that it may be possible to generalize the result to general Gorenstein algebras. 
 We saw that the Gorenstein algebras R/I with fixed  Hilbert function form an irreducible family by <cit.>, so by “general Gorenstein algebra” we refer to a general element in this family. 

In this section we want to compute the codimension for the non-Lefschetz locus of conics _A of a general Gorenstein algebra, fixing the Hilbert function. We will show that the condition on the g-vector is necessary to get the expected codimension if h_⌊e/2⌋-1≠ h_⌊e/2⌋+1.


By Proposition <ref>, we know C_A=_A,⌊e/2⌋-1, so the non-Lefschetz locus of conics has expected codimension 
    _A = min{ h_⌊e/2⌋+1-h_⌊e/2⌋-1, 6}.

By <cit.>
we know that a general Gorenstein algebra has the Strong Lefschetz Property; in particular there exists a linear form ℓ such that ×ℓ^2: [A]_i→ [A]_i+2 has maximum rank for each i, and so ℓ^2∉_A, for A general. As a consequence, we know that for a general Gorenstein algebra A we have _A≥ 1.
As a consequence if h_⌊e/2⌋-1=h_⌊e/2⌋+1 the non-Lefschetz locus must have expected codimension.

    This implies that if A is a general Gorenstein algebra with Hilbert function (1,3,h_2,…,h_e) and socle degree e even, the non-Lefschetz locus always has the expected codimension: it is a hypersurface in ^5 of degree h_e/2-1.




	Let (1,3,h_2,…,h_e) be an SI-sequence such that the positive part of its first difference (1,2,g_2,…,g_⌊e/2⌋) is of decreasing type.
	Then for a general Gorenstein algebra A with Hilbert function (1,3,h_2,…,h_e) the non-Lefschetz locus of conics has expected codimension in ^5,
	
    _A = min{ h_⌊e/2⌋+1-h_⌊e/2⌋-1, 6}.



    If e is even, the _A has expected codimension by <ref>, so we can assume  e is odd.
    
    By semicontinuity, it is enough to construct a Gorenstein algebra A with Hilbert function (1,3,h_2,…,h_e) and non-Lefschetz locus of conics with the expected codimension, and we can proceed exactly as in the proof of Theorem <ref>.
    
	Let R/J be a Gorenstein algebra with Hilbert function (1,3,h_2,…,h_e) that comes from points, so that it is a quotient of R/I_Z, where I_Z is an ideal associated to a set of points  Z with Hilbert function 
    (1,3,h_2,…,h_e-1/2,h_e-1/2… ).
 Since the first difference of the Hilbert function (1,2,g_2,…,g_⌊e/2) is of decreasing type, we can assume Z has the UPP by <cit.>.
    Then a conic C is a Lefschetz conic for R/I if and only if  
    × C: [R/I_Z]_e-1/2-1→ [R/I_Z]_e+1/2
 is injective.
    With the same notation as in the proof of Theorem <ref>, let I_Y=(I_Z:C) be the ideal associated to the set of points Y of Z that C does not pass through. The map × C is injective if and only if [I_Y/I_Z]_e-1/2-1=0. Since Z has the UPP, this happens if and only if C passes through at least
 
    h_e-1/2-h_e-1/2-1+1=h_e-1/2-h_e+1/2-1+1

 points of Z. Since C is a conic, it can not vanish at more than 5 points of Z.
 So we obtain that 
 
    _R/I = min{ h_e+1/2-h_e-1/2-1+1, 6}.



Without the condition on the g-vector, the expected codimension is not necessarily achieved. In fact, if the g-vector of A is not of decreasing type, then _A=1. The proof proceeds in the same way that has been done for lines in <cit.>, and we include it for completeness.



Let A be a general Gorenstein algebra A with Hilbert function (1,3,…, h_e) such that the g-vector of A is not of decreasing type. Then the
non-Lefschetz locus of conics has codimension 1 in ^5.


     Since the g-vector of A, (1,2,g_2,…,g_⌊e/2⌋),  is not of decreasing type, we can find i<⌊e/2⌋
    such that g_i-2>g_i-1=g_i. By <cit.> the generators of I of degree ≤ i have a common factor f and f=g_i. Moreover, g_i-2>g_i-1 and A is general, so the generators of degrees less than i of (I f) span the ideal of a set of points Z in ^2, that we can assume to be reduced <cit.>.
    
    Let C be a form of degree 2 and consider the multiplication map × C: [A]_i-2→ [A]_i. Since i<e-1/2, this map has maximal rank if and only if it is injective. Note that 
    
    [A]_j=[R/I]_j=[R/(I f)· f]_j=[R/(I_Z)· f]_j
 for any j≤ i. So we can consider the map × C: [R/(I_Z)· f]_i-2→ [R/(I_Z)· f]_i. Let Y be the set of points defined by the ideal I_Z C. We have the sequence
    
    0 →[I_Y· f/I_Z· f]_i-2→[R/I_Z· f]_i-2× C→[R/I_Z· f]_i

    and the map × C is injective if and only if [I_Y· f/I_Z· f]_i-2=0. Since the Hilbert function of R/I_Z reaches its multiplicity at i-g_i-2 by <cit.>, it is enough that C passes through one of the points of Z to get that [I_Y· f/I_Z· f]_i-2≠0. So C_A,i-2=1, and therefore C_A=1.

Without the hypothesis of generality, we do not know if there exists a Lefschetz conic, so the codimension of _A could be zero. In fact, for Gorenstein algebras k[x_1,x_2,x_3]/I, even the WLP is an open question. 




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