Relative Richardson Varieties
Melody Chan Nathan Pflueger









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M. Chan]Melody ChanDepartment of Mathematics, Brown University, Box
1917, Providence, RI 02912melody_chan@brown.edu

N. Pflueger]Nathan PfluegerDepartment of Mathematics and Statistics, Amherst College, Amherst, MA 01002npflueger@amherst.edu






  A Richardson variety in a flag variety is an intersection of two Schubert varieties defined by transverse flags.  We define and study relative Richardson varieties, which are defined over a base scheme with a vector bundle and two flags. To do so, we generalize transversality of flags to a relative notion, versality, that allows the flags to be non-transverse over some fibers. Relative Richardson varieties share many of the geometric properties of Richardson varieties. We generalize several geometric and cohomological facts about Richardson varieties to relative Richardson varieties. We also prove that the local geometry of a relative Richardson variety is governed, in a precise sense, by the two intersecting Schubert varieties, giving a generalization, in the flag variety case, of a theorem of Knutson-Woo-Yong; we also generalize this result to intersections of arbitrarily many relative Schubert varieties.  We give an application to Brill-Noether varieties on elliptic curves, and a conjectural generalization to higher genus curves.
 

Relative Richardson varieties
    [
    December 30, 2023
=============================





§ INTRODUCTION
 

A Richardson variety is an intersection of two Schubert varieties defined with respect to transverse flags in a vector space. Here we are concerned with Schubert varieties in Grassmannians and flag varieties.  A very simple example is the subvariety of the Grassmannian 𝔾(1,3) parametrizing lines in ^3 that meet two fixed skew lines.  

Richardson varieties are well-known to be rational, normal, and Cohen-Macaulay, and to have rational singularities, hence they have Euler characteristic 1.  Moreover, the singularities of a Richardson variety are governed entirely by the singularities of the two Schubert varieties: Knutson, Woo and Yong show that the singular points of a Richardson variety are exactly the points that are singular in either one of the Schubert varieties <cit.>. 

In this article we generalize all of these basic results to a relative context. Throughout this paper, we fix an algebraically closed field k of any characteristic. All schemes are assumed to be finite-type over k, and by a “point” of a scheme we will always mean a closed point.
We consider ℓ flag bundles within a vector bundle on a base scheme S, and we allow the flag bundles to become nontransverse over some points of S in a controlled manner: subject to the condition of versality.  



Given a rank-d vector bundle  over a base scheme S, write () for the frame bundle of  and (d) for the variety of complete flags in k^d. Then we define complete flag bundles _1,…,_ℓ  to be versal if the induced map ()→(d)^ℓ is a smooth morphism.  Versality usefully generalizes transversality to a relative context such that properties enjoyed locally by transverse intersections are still enjoyed by versal intersections. But versality is more general than transversality in every fiber.  

A simple example is a 1-parameter family of two complete flags in ^3 which are transverse except over a reduced point p, where the two 2-dimensional subspaces (lines L_1, L_2 in ^3) meet at a point rather than being skew. At each point of this family, one may consider the parameter space of lines in ^3 meeting both L_1 and L_2. These parameter spaces form what we will call a relative Richardson variety, which is in this case a family of smooth quadric surfaces (parameterizing lines through two fixed skew lines) degenerating to a transverse pair of planes over the special point p (one plane parameterizes lines through the intersection point of the fixed lines, while the other parameterizes lines coplanar with the two fixed lines). We now describe the general situation.

Suppose  is a rank d vector bundle over a base scheme S, and  is a complete flag of subbundles.  Fix a nest of sets

    = ({0,…,d-1} = A^i_0⊃ A^i_1⊃⋯⊃ A^i_s = ∅)

in which |A^i_j| = d-i_j.
Let π(i_0,…,i_s;)→ S denote the relative partial flag variety, equipped with tautological flag bundle  inside the rank d vector bundle π^* .
Define the S-scheme X_() to be the subscheme 

    {x∈(i_0,…,i_s;):  (^i_j)_x ∩ (π^* ^a)_x ≥#{a'∈ A^i_j : a'≥ a} for all j,a },

with scheme structure from its description as a degeneracy locus in the usual way, as recalled in <ref>.
The first main theorem is as follows. The proof appears at the end of Section <ref>.
 
Let S be a smooth irreducible k-scheme, let  be a rank-d vector bundle on S, and let ^,^ be a versal pair of complete flags in . 
Let , be nests of sets as in Equation (<ref>). 




  * The relative Richardson variety 
    = X_() ∩ X_()
 is normal and Cohen-Macaulay of pure codimension (ωσ) + (ωτ) in the partial flag variety (i_0,…,i_s;).
Here  is the inversion number, ω is the descending permutation, and the permutations σ=σ(), τ =σ() are the decreasing completions of  and , as defined in <ref>.

  * Letting S' denote the scheme-theoretic image of  in S, the morphism π→ S' satisfies

    π_∗_   =   _S',  and
    
    R^i π_∗_   =    0  for all i > 0.



Moreover, the scheme-theoretic image S' is exactly understood.  It is the subscheme D_τ⋆σ^-1 (;) of S where  and  meet with permutation bounded above, in Bruhat order, by the Demazure product τ⋆σ^-1. See Theorem <ref> for details, and Fact <ref> for recollections on the Demazure product.


 
With the hypotheses of Theorem <ref>,

    H^i(,_) ≅ H^i(S', _S'),

and in particular

    χ( , _) = χ(S', _S').




When the base S is k, a versal pair of flags is the same as a transverse pair of flags in a fixed vector space, and Theorem <ref> directly generalizes several facts about the geometry and cohomology of Richardson varieties. Corollary <ref> generalizes the fact that Richardson varieties have algebraic Euler characteristic 1.



We will say that the map → S' is a cohomological equivalence. Our analysis of the cohomological properties of this map is analogous in several ways to the results in Section 1 of <cit.>.
Indeed, the variety Ω_𝐩,𝐪 defined in <cit.> is a special case of a relative Richardson variety, namely in the case where the flag variety is a Grassmannian, and W_𝐩,𝐪 is its image. The hypotheses in <cit.> are weaker; versality is not required. Under their hypotheses, they prove a K-theoretic equivalence statement, weaker than cohomological equivalence but stronger than equality of Euler characteristic. This allows <cit.> to give an independent proof of the main result of <cit.>, which we prove there using the cohomological equivalence results of this paper.


We can also describe the smooth locus of , as

    ()^sm = (X_)^sm∩ (X_)^sm.

In fact, our second main theorem generalizes this to intersections of arbitrarily many relative Schubert varieties and proves a much stronger result about the singularities of such an intersection.  The most general statement is in Theorem <ref>, and applies to ℓ-fold intersections of degeneracy loci defined with respect to versal flags.  Applied to relative Schubert varieties, we obtain the following special case of Theorem <ref>. The proof appears at the end of Section <ref>.

 
Let P be an étale-local property of finite-type k-schemes that is preserved by products with affine space. Suppose that there is an integer ℓ and a function f_P,ℓ such that for any finite-type k-schemes X_1, ⋯, X_ℓ and point x ∈∏ X_i,

    P( x, ∏ X_i ) = f_P,ℓ( P(π_1(x),X_1), ⋯ P(π_ℓ(x), X_ℓ) ).


Let  be versal complete flags on a scheme S, and _1,⋯,_ℓ be nests of sets, each with the same coranks. Then for every point x ∈ X__1(_1) ∩⋯∩ X__ℓ(_ℓ),

    P(x, X__1(_1) ∩⋯∩ X__ℓ(_ℓ)) = f_P,ℓ( P(x, X__1(_ℓ)), ⋯, P(x, X__ℓ(_ℓ)) ).



This theorem generalizes the flag variety case of a theorem of Knutson, Woo, and Yong <cit.>, both to the relative setting and to ℓ≥ 3. Note that the results of <cit.> apply to general Schubert varieties, whereas our results are specific to Schubert varieties in flag varieties.


We show in Example <ref> that a triple of fixed flags is never versal except in trivial cases. Therefore Theorem <ref> does not apply to intersections of three Schubert varieties in a fixed flag variety; the generalization to ℓ≥ 3 depends in an essential way on the relative context.




 §.§ Application to Brill-Noether varieties
  Relative Richardson varieties arise naturally in the study of Brill-Noether varieties in <cit.>. Let E be an elliptic curve; we will realize twice-pointed Brill-Noether varieties (E,p,q) → S=^d(E) as relative Richardson varieties.  

The schemes (E,p,q) are relative Richardson varieties.

(See Corollary 6.2.)  These varieties are the main building block in the proof in <cit.> of an Euler characteristic formula for Brill-Noether varieties, which uses limit linear series and degenerations of genus g curves to chains of elliptic curves.   Corollary <ref> is used to deduce the Euler characteristics of these building blocks.

In addition, by recognizing twice-pointed Brill-Noether varieties as relative Richardson varieties, we obtain another application: Equation <ref> gives a new proof of the main result of <cit.> in the case of elliptic curves, characterizing the singular locus of (E,p,q).  Previously, the description of the singular locus of twice-pointed Brill-Noether varieties for elliptic curves was obtained in <cit.> using an explicit analysis of vertical and horizontal tangent spaces at points in (E,p,q) relative to the map to the Picard variety ^d(E).

We conjecture that Brill-Noether varieties of a twice-marked curve C of higher genus are also isomorphic as ^d(C)-schemes to relative Richardson varieties; we discuss this conjecture and its consequences in Section <ref>.



Acknowledgments.   We are grateful to Dave Anderson, Allen Knutson, Alex Woo, and Alex Yong for their correspondence regarding an earlier version of this manuscript, in particular for explaining the connection to the Demazure product. We also thank Jonathan Wise for helpful conversations leading to Remark <ref>. MC was supported by NSF DMS-1701924,  NSF CAREER DMS-1844768, and a 2018 Sloan Research Fellowship.



§ NOTATION


We collect for convenience the notation used throughout this paper. Where applicable, page numbers where the notation is first discussed are provided.



rlr
S,T,F     Finite-type k-schemes

     A vector bundle

d     The rank of 

[d]     The set {0,1,⋯, d-1}

, ,     Flags in 

^a     The stratum of codimension a in 

M(H; P_1^, ⋯, P_ℓ^)     Space of relative first-order deformations of ℓ flags     p. loc:mdefn

δ_x(; ^_1, ⋯, ^_ℓ)     Induced map T_x S → M     p. loc:deltadef

A^, B^     Nests of sets     p. loc:nest

σ,τ     Permutations of [d]

σ(A^)     Decreasing completion of a nest of sets     p. loc:dc


r^σ(a,b)     Rank function of a permutation     p. loc:rs

τ⋆σ     Demazure product of permutations     p. fact:demazure

()     Complete (relative) flag variety of a vector bundle 

(i_0, i_1, ⋯, i_s; )     Flag variety of  with strata of codimensions {i_j} 

(d)     Variety of complete flags in k^d 

D_σ(; )     Degeneracy locus where  meets  as prescribed by σ     p. def:dst

D_(; )     Intersection of ℓ degeneracy loci     p. loc:dss

X_()    (Relative) Schubert variety defined by a nest of sets     p. loc:xa

X_σ()     (Relative) Schubert variety defined by a permutation     p. loc:xs

R_,(,)     Relative Richardson variety defined by two nests of sets     p. loc:rab

R_σ,τ(,)     Relative Richardson variety defined by two permutations    p. loc:rst





In all of these notations, we often omit the arguments (e.g. write simply ) where they are clear from context. A tilde over a symbol (e.g., D_σ) indicates the open locus where the defining inequalities of the object in question hold with equality.


§ PRELIMINARIES


This section summarizes background material needed for this paper. General references for this material include the expository article <cit.> and book <cit.> for the Schubert varieties in flag varieties and the Bruhat order, <cit.> for a nice explanation, with many pictures, of rank functions and their relationship to permutations, and <cit.> for degeneracy loci of flags of vector bundles, as well as the essential set of a permutation. We also refer the reader to <cit.> for the Demazure product, as it relates to Schubert varieties. In several cases we use different notation conventions than these sources, more natural to our application; we explain these choices in this section.



 §.§ Permutations and nested sequences
  We begin with combinatorial conventions. We write [d] = {0,…,d-1} and write S_d for the permutation group of [d]. Given a permutation σ∈ S_d, we will write σ = (σ_0,…,σ_d-1) in one-line notation, i.e. σ_i = σ(i).  The inversion number of 
 a permutation σ∈ S_d is
 
    inv(σ)=#{(i,j)∈ [d]^2 |  i<j and σ_i > σ_j}.

We will denote by ω the descending permutation ω(i) = d-1-i. Observe that for all σ∈ S_d, (ωσ) = d2 - (σ), the number of “non-inversions” of σ.

A nest of sets  is a sequence

    A^ = ([d] = A^i_0⊃ A^i_1⊃⋯⊃ A^i_s = ∅),

where |A_i_j| = d - i_j. The numbers i_s are called the coranks of . Note that we require i_0 = 0 and i_s = d for convenience later.

Define the decreasing completion  σ(A^) ∈ S_d of A^ to be the permutation obtained by writing the elements of A^i_0∖ A^i_1 in decreasing order, then the elements of A^i_1∖ A^i_2 in decreasing order, and so on.  
For example, the decreasing completion of

    {0,1,2,3,4}⊃{0,1,3}⊃∅

is (4,2,3,1,0). 
For a nest of sets , we define inv() = inv(σ()).
Decreasing completion provides a bijection between S_d and nests of sets of coranks (0,1,⋯,d-1). Such a nest  is called complete. We will often identify complete nests of sets with permutations.



 §.§ Flags
 


Let H be a d-dimensional vector space over k.  Write 

    P^ = (P^0 ⊃ P^1 ⊃⋯⊃ P^d = 0)

for a complete flag of subspaces of H, where P^i has codimension i.  The relative position of two flags  and   uniquely defines a rank function

    r(a,b) =  P^a ∩ Q^b.


The rank function of two complete flags can be encoded by a permutation. For any σ∈ S_d, define the rank function of σ by

    r^σ(a,b) = #{a' ∈ [d]: a' ≥ a  and σ(a') ≥ b }.

Note that this notation does not exclude the cases a ≥ d or b ≥ d, where we define r^σ(a,b) = 0.

  For any two complete flags ,, there exists a unique σ∈ S_d such that P^a ∩ Q^b = r^σ(a,b) for all a,b ∈ [d], called the permutation associated to ,. The following are equivalent.


  * The permutation associated to , is σ.

  * There exists a basis v_0, v_1, ⋯, v_d-1 of H such that {v_a, ⋯, v_d-1} is a basis for P^a for all a, and {v_σ(b), ⋯, v_σ(d-1)} is a basis for Q^b for all b.
 

For example, = if and only if σ = id.  At the other extreme, flags P^,Q^ are transverse if their associated permutation is ω. Explicitly, , are transverse if and only if 

    P^a ∩ Q^b = max(d - a - b, 0)

for all i, j; that is, every pair of subspaces meets transversely.
Call , almost-transverse if their associated permutation differs from ω by an adjacent transposition, or equivalently (ωσ) = 1.

A rank function r^σ is uniquely determined by its values on a fairly small subset of its domain.


The essential set of a permutation σ∈ S_d is

    (σ) = { (a,b): 1 ≤ a,b < d, σ(a-1) < b ≤σ(a) σ^-1(b-1) < a ≤σ^-1(b)}.





The essential set was introduced in <cit.>, although we define it slightly differently here; see Remark <ref>.
The importance of the essential set is reviewed in Fact <ref>. 

The set S_d has a partial order, the Bruhat order: σ≤τ in Bruhat order if and only if r^σ(a,b) ≥ r^τ(a,b) for all a and b.  See, e.g., <cit.>. By semicontinuity, the associated permutation of two varying flags is lower semi-continuous in the Bruhat order.



Let P^ denote the vector subspace of H consisting of ϕ: H → H such that ϕ(P^i) ⊆ P^i for all i.   
The following characterization of (σ), for a permutation associated to flags ,, will be convenient later.


 
If σ is the permutation associated to flags P^, Q^, then 

    (ωσ) =  H - (  P^ +  Q^).

In particular, P^ and Q^ are transverse if and only if P^ +  Q^ =  H.


Fact <ref> can be proved using a straightforward argument characterizing P^∩ Q^ in terms of a basis of the type described in Fact <ref>.







 §.§ The Demazure product


To state our main results, we require an associative operation ⋆ on S_d called the Demazure product.

 
For any two permutation σ, τ∈ S_d, there exists a unique permutation τ⋆σ such that

    r^τ⋆σ(a,b) = max_0 ≤ k ≤ d( r^σ(a,k) + r^τ(k,b) - (d-k) ).

The operation ⋆ defined in this way is associative, and satisfies (σ⋆τ)^-1 = τ^-1⋆σ^-1. When one of the permutations is a simple transposition s (a transposition of two adjacent elements of [d]), then 

    τ⋆ s = τ    if (τ s) < (τ) 
    τ s    if (τ s) > (τ).



Equation <ref> is motivated by the following observation: if P^, Q^, R^ are three flags, σ is the permutation associated to P^, Q^, and τ is the permutation associated to Q^, R^, then for all a,b,k ∈ [d],

    P^a ∩ R^b ≥ P^a ∩ Q^k +  Q^k ∩ R^b -  Q^k = r^σ(a,k) + r^τ(k,b) - (d-k).

Therefore τ⋆σ gives an upper bound on the permutation associated to P^, R^. In fact, one can deduce from Theorem <ref> that τ⋆σ is the minimal such permutation. 

The Demazure product was introduced and studied in <cit.> and <cit.> for arbitrary Weyl groups. We briefly sketch a proof of Fact <ref> for the benefit of the reader unfamiliar with these topics, as follows. One may use the right side of Equation <ref> to define an operation on functions [d] × [d] → [d]. One can verify that this operation is associative by writing a composition of two products as a maximum taken over two variables, and Equation <ref> may be verified, on the level of rank functions, by some casework. Finally, the existence of a permutation with the desired rank function may be proved by induction on (σ) by factoring σ into simple transpositions. 








 §.§ Schubert varieties in flag varieties
 

Fix a vector space H of dimension d, and let F^∈(H) be a fixed complete flag. Given σ∈ S_d, define the Schubert variety X_σ by

    X_σ = X_σ(F^) = {∈(H) V^a∩ F^b≥ r^σ(a,b)  a,b ∈ [d]}.


We write X_σ for the open locus where all these defining inequalities hold with equality.


Our conventions differ from those used in <cit.> and elsewhere, since we index our flags by codimension, rather than dimension. We choose this convention because it is most natural for our application in <cit.>, where we stratify sections of a line bundle by their vanishing order at a point.
For example, in <cit.>, the associated permutation w of two flags V_,W_ (indexed by dimension) is defined by V_a ∩ W_b = r_w(a,b), where w is a permutation of {1,2,⋯,d} and the rank function is r_w(a,b) = #{i ≤ a: w(i) ≤ b}. There are two ways to translate our notation to the notation of <cit.>.



  * Define an isomorphism i: (H) →(H^∨) by i() = V_, where V_a = (P^a)^⊥. If i() = V_ and i() = W_, then 

    V_a ∩ W_b = a+b-d +  P^a ∩ Q^b,

from which it follows that P^a ∩ Q^b = r^σ(a,b) if and only if V_a ∩ W_b = r_w(a,b), where w(i) = σ(i-1)+1 (in one-line notation, w is obtained by adding one to all entries of σ). So i(X_σ) is equal to the Schubert variety denoted X_w in <cit.>.

  * Define V_a = P^d-a and W_b = Q^d-b. If P^a ∩ Q^b = r^σ(a,b) for all a,b, then V_a ∩ W_b = r_w(a,b), where w(i) = d- σ(d-i) (in one-line notation, w is obtained by adding 1 to all entries of ωσω). So our X_σ is equal to the variety denoted X_w in <cit.>.






We collect facts about X_σ. It is well-known that X_σ is irreducible, normal, and Cohen-Macaulay, of codimension (ωσ) in (H). A criterion for whether X_σ is regular is given by Lakshmibai and Sandhya <cit.>: X_σ is regular if and only if σ is a 3120 and 2301-avoiding permutation.[In standard notation, 4231 and 3412-avoiding.] 
The singular locus of X_σ is closed and a union of Borel orbits; therefore, it must be a union of varieties X_σ' for 
σ'≤σ.
Lakshmibai-Sandhya conjectured a combinatorial description of which Schubert subvarieties X_σ' occur, and their conjecture was proven independently by several groups <cit.>. The description shows that σ' ranges over all permutations that are derived from minimal 3120 and 2301 patterns in σ by a certain combinatorial modification, see e.g., <cit.>.  We note that the singular locus of X_σ is a union of X_σ' for 
σ'<σ 
ranging over a certain set of permutations having at least two fewer inversions than σ.  In particular, X_σ is regular in codimension 1.


More generally, if  is a nest of sets of coranks 0 = i_0 < ⋯ < i_s = d, then we define a Schubert variety in the partial flag variety (i_0, ⋯, i_s; H) as follows.

    X_(F^) = {∈(i_0, ⋯, i_s; H):  V^i_j∩ F^b ≥ r^σ()(i_j,b)  j ∈ [s], b ∈ [d] }.

We write X_ for the open locus where these defining inequalities hold with equality. The inverse image of X_ under the forgetful map (H) →(i_0, ⋯, i_s; H) is equal to X_σ(F^). Since this forgetful map is a fiber bundle with smooth irreducible fibers, most of the geometric facts above carry over readily to X_.









 §.§ Degeneracy loci and relative Schubert varieties

We adopt the following notation convention. If , are two sub-bundles of a vector bundle  on a scheme S, we will write

    {x ∈ S: _x ∩_x ≥ r}

as a shorthand for the subscheme defined by the degeneracy locus where the bundle map →/ has rank at most rank() - r, defined locally as a determinantal variety in the usual way (e.g. as in <cit.> or <cit.>). In particular, we always mean this notation as a scheme-theoretic definition. 

We will be concerned with degeneracy loci of the following form. For  a rank d vector bundle on a scheme S, and ^, ^ complete flags in , we consider the subscheme

    {x ∈ S:  ( ^a )_x ∩ (^b)_x 
    ≥ 
    r^σ(a,b)  a,b ∈ [d]}.


In fact, many of the inequalities in this definition are redundant.

  (<cit.>)
The scheme described by Equation (<ref>) is equal to the scheme

    { x ∈ S:  ( ^a )_x ∩ (^b)_x 
    ≥ 
    r^σ(a,b)  (a,b) ∈(σ)}.



 
The definition of the essential set in <cit.> is different from ours, because the degeneracy loci under consideration are defined by rank(E_p → F_q) ≤ r_w(q,p), where E_1 ↪ E_2 ↪⋯ E_n and F_n ↠ F_n-1↠⋯↠ F_1 are vector bundles indexed by rank. The essential set of <cit.> is the set of (q,p) for which the condition rank(E_p → F_q) ≤ r_w(q,p) is essential. Our definition is obtained by a straightforward translation. 


In light of Fact <ref>, we make the following definition, which allows for partial flags.

 
Let  a rank d vector bundle on a scheme S, and ^ = (^i_0⊃⋯⊃^i_s), ^ = (^j_0⊃⋯⊃^j_t ) be flags in . Let σ be any permutation such that (σ) ⊆{i_0, ⋯, i_s}×{j_0, ⋯, j_t}. Define a subscheme 

    D_σ(^; ^) = {x ∈ S:  ( ^a )_x ∩ (^b)_x 
    ≥ 
    r^σ(a,b)  a,b ∈(σ) }.

When , are complete flags, also define D_σ(^;^) to be the open subscheme where we have equality (^a)_x ∩ (^b)_x = r^σ(a,b) for all (a,b) ∈ [d]^2 (not only those in (σ)). Thus D_σ(^; ^) is the locus where the two flags have associated permutation σ, and D_σ(^; ^) is the locus where the two flags have associated permutation 

at most
σ in Bruhat order.

 
Suppose that  is complete, and the strata of  have coranks 0 = i_0 < ⋯ < i_s = d. Then the following is a useful sufficient condition for D_σ(; ) to be well-defined: for all 0 ≤ j < s, σ(i_j) < σ(i_j + 1) < ⋯ < σ(i_j+1-1). This condition ensures that σ(a-1) > σ(a) for all a except possibly when a ∈{i_0, ⋯, i_s} and thus (σ) ⊆{i_0, ⋯, i_s}× [d]. In other words, if  is any nest of sets with coranks i_0, ⋯, i_s, then D_σ()(; ) is well-defined. 


We mention some important geometric facts about degeneracy loci, stated at the level of generality we need; see <cit.> for the more general statement, including an intersection theory result.

 (<cit.>) 
Suppose S is Cohen-Macaulay and pure dimensional. Then any component of (; ) has codimension at most (ωσ). If the codimension of  is exactly (ωσ), then  is Cohen-Macaulay.


Given ℓ+1 flags , _1,…,_ℓ and ℓ permutations σ_1,…,σ_ℓ, we also use the following abbreviation.

    D_σ_1,…,σ_ℓ(; _1,…,_ℓ) = D_σ_1(; _1) ∩⋯∩ D_σ_ℓ(; _ℓ).

When the flags are clear from context, we will omit the arguments and write simply D_σ or D_σ_1,…,σ_ℓ.


 
It is sometimes convenient to view D_σ locally as the inverse image of a Schubert variety. If U ⊂ S is an open subscheme on which  is trivial, 
then we may 
choose completions of  and  
and locally trivialize  
in a way that makes the completion of  constant. Then the completion of  defines a morphism p: S →(d) (under which the completion of  is the pullback of the tautological bundle), and we have, scheme-theoretically,

    D_σ(; ) = p^-1 (X_σ).



 (Relative Schubert varieties)
Let S be a scheme,  a rank-d vector bundle on S, and ^ a complete flag in . Let π: () → S denote the relative flag variety. For every σ∈ S_d, there is a relative Schubert subvariety X_σ() ⊆(). These subvarieties are important special cases of the degeneracy loci defined above, namely

    X_σ() = D_σ(; π^∗^)

where  is the tautological flag bundle. Similarly, we obtain relative Schubert varieties in partial flag varieties:

    X_() = D_σ()(; π^∗).





§ VERSALITY


In this subsection, we define versality of complete flags, and prove several criteria for it. We work exclusively with complete flags in this subsection; results for incomplete flags can be deduced from the case of complete flags.

Let  be a vector bundle of rank d on a base scheme S.
Denote by (d)=(k^d) the variety of complete flags in the standard vector space k^d, and denote by () → S the frame bundle of . Then a complete flag ^ in  uniquely determines a morphism of schemes () →(d). 

 
With the notation above, suppose that _1, ⋯, _ℓ are complete flags in , inducing a morphism

    p: () →(d)^ℓ.

Call the ℓ-tuple of flags (_1, ⋯, _ℓ) versal if p is a smooth morphism. Call the flags versal at x ∈ S if they are versal when restricted to some neighborhood of x.


Observe that any subset of a versal ℓ-tuple of flags is again versal, since the projection (d)^ℓ→(d)^ℓ' is smooth for all ℓ' < ℓ.

When ℓ=1, versality is automatic. When ℓ = 2 and S =  k, two flags are versal if and only if they are transverse, as is explained in Example <ref>. When ℓ = 2 but S is a more general scheme, Definition <ref> is equivalent to a geometric condition (Lemma <ref>) that roughly says that the locus where ^, ^ are nontransverse is stratified by smooth varieties of specific codimension. This stratification is indexed by permutations. 

Our first goal is a linear-algebraic criterion for versality (Proposition <ref>), for which we need some preliminary notions.
For complete flags P^_1, ⋯, P^_ℓ in a vector space H, let 


    M=M(H; P^_1, ⋯, P^_ℓ)=coker ( H ∏_i=1^ℓ H /  P^_i),

where 
Δ is the diagonal map.

This vector space M is the space of relative first-order deformations of the flags.  Indeed, each factor H /  P^_i is naturally identified with the tangent space at [P^_i] to the flag variety, 
while the image of Δ corresponds to simultaneous deformations arising from a change of basis for H. 

Now given x∈ S, let 

    δ_x = δ_x(; ^_1, ⋯, ^_ℓ): T_x S → M(_x; (P^_1)_x, ⋯, (P^_ℓ)_x)

denote the natural linear map encoding the first-order deformations of the P^_i relative to each other induced by a first-order deformation in S. A precise definition is as follows.
Note the natural map p()→(d)^ℓ induced by the flags _1^,…, _ℓ^.  Shrink S so that  is trivial.  Any section s S→() of ()→ S induces a linear map

    δ_x T_x S ⟶ T_(p∘ s)(x)(d)^ℓ⟶ M,

where the first arrow is the differential of p∘ s at x, and the second is induced by s(x).

Moreover, any two sections s,s' are related by an element of GL_d(S). That is, there is a morphism c S→GL_d with the following commuting diagram.

    S@=[d] [r]^-c× s   GL_d ×() [d] [r]    GL_d ×(d)^ℓ[d] 
    
     S [r]^-s'   () [r]    (d)^ℓ

Passing to tangent spaces verifies that δ_x does not depend on choice of s.








 
Let S be a scheme with a vector bundle  and complete flags ^_1, ⋯, ^_ℓ. For any x ∈ S, these flags are versal at x if and only if S is smooth at x and δ_x is surjective.



Fix x ∈ S and a point y ∈() in the fiber over x. First, we claim that the map δ_x is surjective if and only if the differential dp_y of the map p() →(d)^ℓ is surjective. Shrinking S if necessary, choose a section s: S →() such that y = s(x). We use the description of δ_x in Equation <ref>. The kernel of the map T_(p ∘ s)(x)(d)^ℓ→ M is equal to the tangent space to the _d-orbit of (p ∘ s)(x). Since p is equivariant, this is the image under dp_s(x) of the _d-orbit of s(x). Therefore the map T_s(x)() → M is surjective if and only if dp_s(x) is surjective. Furthermore, the _d-orbit of s(x) is the fiber of x in (), so its tangent space is complementary to the image of ds_x. Therefore the image of δ_x is equal to the image of T_s(x)() → M. Putting this together, δ_x is surjective if and only if dp_s(x) is surjective.

Next, observe that since () is a _d-torsor, it is smooth at y if and only if S is smooth at x.

Suppose that the flags are versal at x. Since (d)^ℓ is a nonsingular variety, the structure map (d)^ℓ→ k is smooth, hence the composition () → k is smooth at y, i.e., y is a smooth point of () and x is a smooth point of S. Since p is a smooth morphism of nonsingular varieties in a neighborhood of y, the differential dp_y is surjective <cit.>. It follows that δ_x is surjective as well.

Now suppose x ∈ S is a smooth point and δ_x is surjective. Then y is a smooth point of () and dp_y is surjective, so p is a morphism of nonsingular varieties with surjective differential around y. Hence p smooth at y, and the flags are versal at x.



 
The definition of versality and the criterion of Proposition <ref> has a stack-theoretic description, as follows. A choice of ℓ complete flags on S is equivalent to a morphism p from S to the quotient stack [ (d)^ℓ / _d ], which may be regarded as the moduli stack of ℓ-tuples of flags. This morphism, along with the induced morphism p: () →(d)^ℓ discussed above, form a cartesian diagram as follows.

    ()[r]^-p[d]    (d)^ℓ[d] 
    
    S [r]^-p   [ (d)^ℓ / _d ]

The vertical arrows are _d-torsors, and it follows that p is smooth if and only if p is smooth. So the tuple of flags is versal if and only if it determines a smooth morphism to the moduli stack; this accords with the usual use of “versal” in deformation theory. 
The differential of p at x ∈ S may be identified with a map from T_x S to a two-term complex H ∏_i=1^ℓ H /  P^_i, which is surjective if and only if the linear map δ_x is surjective. Hence Proposition <ref> amounts to the fact that p is smooth if and only if it has smooth domain and surjective differential.



 
(Versality of fixed flags)
A pair of two complete flags ^, ^ are versal in a neighborhood of any smooth point x ∈ S where ^_x and ^_x are transverse. This is because P^_x +  Q^_x =  H by Fact <ref>, which implies (and is equivalent to) M(H_x; P_x^, Q_x^) = 0. 
If S= k, the flags are transverse if and only if they are versal. Therefore versality is a generalization of transversality.

When ℓ > 2 and d > 2, then ℓ flags over S= k are never versal since 
    ∏ H/_i = ℓd2≥ 3 d2≥ d^2 =  H,
 so H →∏ H /  P^ cannot be surjective because the kernel always contains the identity and is therefore nontrivial. When d=2, the only way versality over k can occur is if ℓ≤ 3 and the flags are distinct.


Adding a tautological flag bundle preserves versality:

 
With  a vector bundle on a smooth scheme S, if _1,…,_ℓ are versal complete flags on S, and π:() → S is the flag variety of  with tautological bundle ^, then π^∗_1, …,π^∗_ℓ, ^ are versal flags of π^∗ on (). 



Fix x∈ S, and define P^_i = (_i)_x. Shrinking S, we may assume = H× S. For any point y = [] ∈(H), versality at (x,y) ∈() is equivalent to the surjectivity of the linear map

    T_(x,y)() ⊕ H →∏ H/_i × H/.

We have T_(x,y)() = T_x S ⊕ T_y (H).  Then surjectivity of the above map follows from the fact that T_x S ⊕ H →∏ H/_i is surjective by the versality hypothesis, and T_y (H) → H / is an isomorphism.




 The linear maps δ_x provide a description of tangent spaces to degeneracy loci D_σ(; ) defined in Definition <ref>, and intersections thereof.

 Let D_σ(;) be as in Definition <ref>.


  * At any point x ∈ D_σ( ; ),

    T_x D_σ( ; ) ⊇δ_x (; , ).


  * Moreover, if x ∈D_σ(; ), then equality holds.




Start by verifying (2) in the 
special case that H=k^d and S = (H)^2 with , the tautological flag bundles.  

For any x = (,) ∈D_σ( ; ), 
the scheme D_σ( ; ) is equal to the _d-orbit of x, i.e. the scheme-theoretic image of the map _d →(d)^2 taking 1 to x.  The differential of this map is the diagonal map Δ H→ H/× H /,
and T_x D_σ( ; ) = T_x D_σ( ; )= imΔ = δ_x as desired.

Now the general case of (2) follows by pulling back: shrink S around x so that we may assume  is trivial, and then choose a section s S→(H); we get a composite map S→(H)→(d)^2 taking x to y=(,), say.  Under the differential
T_x S → T_y (d)^2,
δ_x is the preimage of δ_y = imΔ, by definition of δ_x. (Note this does not depend on the choice of section, as verified in the definition of δ_x).  And D_σ (;) is the inverse image of D_σ(_1;_2)⊂(d)^2.  This verifies (2) in general.

Now (1) follows from (2) by observing that if x ∈ D_σ(;), then x ∈D_σ' (;) for some 

σ' ≤σ 
in Bruhat order, so since D_σ'(;) is a subscheme of D_σ(;),

    T_x D_σ(; ) ⊇ T_x D_σ' (;) = δ_x(; , ).








 
If x ∈D_σ(;), where , are complete flags, then the following are equivalent.


  * The pair (^, ^) is versal at x.

  * The point x is a smooth point of S and δ_x is surjective.

  * The point x is a smooth point of both S and D_σ(^;^) and the local codimension of D_σ(^;^) in S is equal to (ωσ).




We first observe that in the two-flag case, we have the following isomorphism.

    M(H_x; P^_x, Q^_x) ≅ H_x / (  P^_x +  Q^_x )

Together with Fact <ref> and the assumption that σ is the permutation associated to P_x^,Q^_x, this implies that M(H_x; P^_x, Q^_x) = (ωσ). By Lemma <ref>, we deduce that 

    T_x D_σ(^; ^) =  T_x S - (ωσ) + cokerδ_x.


We now prove the Lemma. The equivalence of (1) and (2) is part of Proposition <ref>, so it suffices to prove that (2) is equivalent to (3). Assume that x is a smooth point of S (since this is a hypothesis of both statements). Observe that

    _x S - (ωσ) ≤_x D_σ(^; ^) ≤ T_x D_σ(^; ^) = _x S - (ωσ) + cokerδ_x.

The first inequality follows from the local description of D_σ(^, ^) as the inverse image of a Schubert variety (Remark <ref>). Now, δ_x is surjective if and only if cokerδ_x = 0, which holds if and only if both inequalities above hold with equality. This in turn is equivalent to D_σ(^; ^) having local codimension (ωσ) (first inequality) and x being a smooth point of it (second inequality). So indeed (2) is equivalent to (3).



A simple example of all conditions in Lemma <ref> being satisfied is two points in ℙ^1 moving above a smooth 1-parameter base S, which come together  over a reduced point x of S.




§ A KNUTSON-WOO-YONG THEOREM FOR DEGENERACY LOCI OF VERSAL FLAGS
 

We turn our attention to intersections of degeneracy loci defined with respect to versal flags. We show that the singularities of these loci are completely controlled by the singularities of the individual degeneracy loci, and in turn by Schubert varieties. More precisely, we prove the following analog of the main theorem of <cit.>. The results of <cit.> concern general Schubert varieties, whereas we are concerned only with Schubert varieties of flag varieties. In the flag variety case, our result provides a generalization to ℓ-fold intersections, as well as to the relative setting.

 
Let P be an étale-local property of finite-type k-schemes that is preserved by products with affine space. Suppose that there is an integer ℓ and a function f_P,ℓ such that for any finite-type k-schemes X_1, ⋯, X_ℓ and point x ∈∏ X_i,

    P( x, ∏ X_i ) = f_P,ℓ( P(π_1(x),X_1), ⋯ P(π_ℓ(x), X_ℓ)).

Let , _1, _2, ⋯, _ℓ be a versal (ℓ+1)-tuple of flags in a rank-d vector bundle  on a smooth variety S, σ_1,⋯,σ_ℓ∈ S_d, and x a point in D_σ_1,⋯,σ_ℓ(; _1,⋯,_ℓ). Then

    P(x,D_σ_1,⋯,σ_ℓ(; _1,⋯,_ℓ)) = f_P,ℓ( P(x, D_σ_1(; _1)), ⋯, P(x, D_σ_ℓ(; _ℓ))).


Throughout this section, assume that we have fixed a property P, denoted P(x,X) for a point x∈ X, and a function f_P,ℓ satisfying the hypotheses of Theorem <ref>.

 
Let X,Y be schemes with points x ∈ X, y ∈ Y. Let ,𝒥 be rank-d vector bundles on X,Y respectively, let _1, ⋯, _ℓ be flags in  and let _1,⋯, _ℓ be flags in 𝒥.

We say that (x,_1 ⋯,_ℓ) is equivalent via smooth morphisms to (y, _1,⋯,_ℓ) if there is a scheme Z with rank-d vector bundle 𝒦, two smooth morphisms π: Z → X, ρ: Z → Y and a point z ∈ Z such that π(z) = x, ρ(z) = y, 𝒦≅π^∗≅ρ^∗𝒥, and such that upon identifying both pullbacks with 𝒦, we have π^∗_i = ρ^∗_i for all i.


Equivalence via smooth morphisms is an equivalence relation. Reflexivity and symmetry are clear, and transitivity follows from standard facts about fiber products of smooth morphisms.

 
If (x, , _1, ⋯, _ℓ) is equivalent via smooth morphisms to (y, , _1, ⋯, _ℓ), and σ_1, ⋯, σ_ℓ are permutations such that 
x ∈ D_σ_1,⋯,σ_ℓ(; _1, ⋯, _ℓ),
then 

    y ∈ D_σ_1,⋯,σ_ℓ(; _1, ⋯, _ℓ)

and

    P(x, D_σ_1,⋯,σ_ℓ(; _1, ⋯, _ℓ)) = P(y, D_σ_1,⋯,σ_ℓ(; _1, ⋯, _ℓ)).

Furthermore, the codimension at y of D_σ_1,⋯,σ_ℓ(; _1, ⋯, _ℓ) in Y is equal to the codimension at x of D_σ_1,⋯,σ_ℓ(; _1, ⋯, _ℓ) in X.



Let π: Z → X, ρ Z→ Y, and z ∈ Z be as in Definition <ref>. For simplicity, abbreviate D_σ_1,⋯,σ_ℓ(; _1,⋯,_ℓ) by D_X, D_σ_1,⋯,σ_ℓ(; _1,⋯,_ℓ) by D_Y, and D_σ_1,⋯,σ_ℓ(π^∗; π^∗_1,⋯,π^∗_ℓ) by D_Z. Observe that D_Z = π^-1 (D_X) = ρ^-1(D_Y). Since x ∈ D_X, it follows that z ∈ D_Z and y ∈ D_Y.

The restriction D_Z → D_X of π is smooth. Let n be the relative dimension of this morphism at z. Then n is also the relative dimension of π at z; it follows that the codimension of D_Z in Z is equal to the codimension of D_X at x. 

There exist affine neighborhoods z ∈ U ⊆ D_Z and x ∈ V ⊆ D_X and an étale morphism e: U →𝔸^n_V such that U → V factors through the projection 𝔸^n_V → V <cit.>. Since the property P is étale-local and unaffected by products with affine space, it follows that P(z,D_Z) = P(e(z), 𝔸^n_V) = P(x, D_X). 

Applying the same logic to ρ, it follows that the codimension of D_Y in Y is equal to the codimension of D_Z in Z, and P(z,D_Z) = P(y,D_Y). The result follows.



The frame bundle () → S is surjective, so x ∈ S lifts to a point x' ∈(). The versal tuple (, _1, ⋯, _ℓ) determines a smooth morphism p: () →(d)^ℓ+1. Next, apply the same construction, with S replaced by (d)^ℓ, with T = k^d ×(d)^ℓ being the trivial rank d vector bundle, and 
with flags (F^, _1,⋯, _ℓ), where {_i} are the tautological flags and F^ is an arbitrary fixed flag.  
We obtain a morphism v: (T) →(d)^ℓ+1. The tuple (F^, _1,⋯, _ℓ) is versal, by ℓ applications of Lemma <ref> (and the straightforward observation that a single flag is versal), so v is smooth. It is straightforward to check that v is also surjective, so p(x') ∈(d)^ℓ+1 lifts to x”∈(T)= _d ×(d)^ℓ. Denote by y the image of x” in (d)^ℓ. Denote by π_1,⋯,π_ℓ the projection maps from (d)^ℓ to (d). The maps and points constructed are summarized in Figure <ref>. In this diagram, all morphisms are smooth.



The intuition behind the first four arrows of the diagram is that a family of ℓ+1 versal flags can, after coordinate change, be regarded as a family of ℓ versal flags together with one fixed flag.

Denote the tautological flags in the trivial bundle on (d)^ℓ+1 by 𝒰^_1,⋯, 𝒰^_ℓ+1. By construction, p^∗𝒰^_1, ⋯, p^∗𝒰^_ℓ+1 are equal to the pullbacks from S of , _1, ⋯, _ℓ, respectively. Similarly, v^∗𝒰^_1, ⋯, v^∗𝒰^_ℓ+1 are equal to the pullbacks from (d)^ℓ+1 of F^, _1, ⋯, _ℓ, respectively. Finally, denoting the tautological and trivial flags over (d) by , E^ respectively, we have _i = π_i^∗ and F^ = π_i^∗ E^. From this, we deduce the following equivalences via smooth morphisms.


    (x, , _1, ⋯, _ℓ)    ∼    (y, F^, _1, ⋯, _ℓ)
    
    (x, , _i)    ∼    (π_i(y), E^, )


Observe that for each i, D_σ_i(F^; _i) = π^-1_i ( D_σ_i(E^; )) inside (d)^ℓ. Therefore

    D_σ_1, ⋯, σ_ℓ(F^; _1, ⋯, _ℓ)    =   ⋂_i π_i^-1 (D_σ_i(E^; ))
       ≅   ∏_i D_σ_i(E^; ).

Putting this together and applying Lemma <ref>:

    P(x, D_σ_1, ⋯, σ_ℓ(; _1, ⋯, _ℓ))    =    P(y, D_σ_1,⋯,σ_ℓ(F^; _1,⋯, _ℓ))
       =    f_P,ℓ(
    P(π_1(y), D_σ_1(E^; )), ⋯, P(π_ℓ(y), D_σ_ℓ(E^; )) 
    )
       =    f_P,ℓ(
    P(x, D_σ_1(; _1)), ⋯, P(x, D_σ_ℓ(; _ℓ))
    ).



In fact, the proof of Theorem <ref> shows that we can say slightly more: 
we can reduce completely to Schubert varieties in flag varieties.

 
With the same hypotheses as Theorem <ref>, let 
σ'_1 ≤σ_1, ⋯, σ'_ℓ≤σ_ℓ 

be permutations such that x ∈D_σ'_i(; _i) for all i. Let z_i be any point in the open Schubert cell X_σ'_i⊆(d). Then

    P(x,D_σ_1, ⋯, σ_ℓ(; _1, ⋯, _ℓ)) = f_P,ℓ( P(z_1, X_σ_1), ⋯, P(z_ℓ, X_σ_ℓ)).




In the notation of the proof of Theorem <ref>, it suffices to prove that 
    P(π_i(y), D_σ_i(E^; )) = P(z_i, X_σ_i).
 
Observe that D_σ_i(E^; ) = D_σ_i^-1(; E^) = X_σ_i^-1(E^). 
Similarly, D_σ_i'(E^,) = X_(σ_i')^-1(E^). Since 
Schubert cells are Borel orbits,
it follows that for any w ∈X_(σ_i')^-1(E^), 

    P(π_i(y), D_σ_i(E^; )) = P(w, X_σ_i^-1).
 
To see that the same result holds when σ_i^-1, (σ_i')^-1 are replaced with σ_i, σ'_i, observe that we may apply the entire argument to the case S = (d), x = w, ℓ = 1,  =  , _1 = E^, from which it follows that for any z_i ∈X_σ'_i, P(w,X_σ_i^-1(E^)) = P(z_i, X_σ_i(E^)), and the result follows.


We also point out another useful consequence of the proof of Theorem <ref>.

 
If , _1, ⋯, _ℓ is a versal (ℓ+1)-tuple of flags in a vector bundle  over S, then
D_σ_1,⋯,σ_ℓ(; _1, ⋯, _ℓ) has pure codimension (ωσ_1) + ⋯ + (ωσ_ℓ) in S.



By the codimension statement in Lemma <ref> and the equivalence via smooth morphisms in the proof of Theorem <ref>, we may assume that S = (d)^ℓ, _i = 𝒰^_i, and = F^. The result now follows from the fact that, in (d), the codimension of D_σ_i(; F^) is equal to the codimension of X_σ_i, which is (ωσ_i) (see Section <ref>).



Compare Corollary <ref> below with <cit.>; note also the generalization from ℓ=2 to any ℓ.

By making various choices of P and  f_P,ℓ, we deduce the following about any degeneracy locus D = D_σ_1, ⋯, σ_ℓ for versal flags ,_1,…,_ℓ.

    
  * The smooth locus of D is the intersection of the smooth loci of each D_σ_i. 
    
  * D is normal and Cohen-Macaulay.
    
  * D is Gorenstein at x∈ D if and only if each X_σ_i is Gorenstein along X_σ_i', where σ_i' is as in the hypotheses of Theorem <ref>. 
    
  * Recall that for x∈ X, the H-polynomial H_x,X(q) is defined by the equation
    
    Hilb(G_𝔪_x(_X,x);q) = H_x,X(q)/(1-q)^_x X,

    where G_𝔪_x(_X,x) denotes the associated graded ring of _X,x.  Recall the Hilbert-Samuel multiplicity is mult_x,X = H_x,X(1).  Then
    
    H_x,D(q) = ∏_i H_x,D_σ_i(q),

    and hence
    mult_x,D = ∏_i  mult_x,D_σ_i.




For (1), (2) and (3), recall that Cohen-Macaulayness, reducedness, 
normalness, regularity, and being Gorenstein are all étale-local <cit.>; moreover all Schubert varieties are normal and Cohen-Macaulay. In all three cases, we take the function f_P,ℓ to be “logical and.”  For (4), the associated graded of _D,x can be computed from its completion, and étale morphisms induce isomorphisms on completed local rings (recall we always work over an algebraically closed field).  Hence the H-polynomial is also an étale-local invariant. The result follows by taking f_P,ℓ to be the usual multiplication of polynomials. 


See <cit.> for a characterization of which Schubert varieties, in type A, are Gorenstein in terms of permutation pattern-avoidance, and a conjectured general characterization of the non-Gorenstein locus. The conjecture is proven by Perrin for minuscule Schubert varieties <cit.>; it is open in general as far as we know.

We can now prove Theorem <ref> from the introduction. This requires a short argument applying Theorem <ref> to cases where one defining flag may not be a complete flag.


Let i_0, ⋯, i_s be the coranks of the nests of sets. Let F = (i_0, ⋯, i_s; ) and F' = (H) (the complete flag variety); denote the projections to S by π, π' respectively. Let  be the tautological complete flag in π'^∗. Since each essential set (σ(_i)) is contained in {i_0, ⋯, i_s}× [d], we see by comparing defining rank conditions that the inverse image of X__1(_1) ∩⋯ X__ℓ(_ℓ) in F' is equal to D_σ(_1), ⋯, σ(_ℓ) ( ; π'^∗_1, ⋯, π'^∗_ℓ), to which Theorem <ref> applies. The Theorem now follows from the fact that the forgetful map F' → F is a fiber bundle, with fibers étale-locally isomorphic to affine space.




§ RELATIVE RICHARDSON VARIETIES





 §.§ Definitions


We now define relative Richardson varieties and deduce their basic properties from the results of the previous section.  In particular we take ℓ=2 in this section, as we are not aware of a generalization of the cohomological arguments below to higher  ℓ. 

 
Let S be an irreducible smooth variety, with a rank d vector bundle  and two complete versal flags ,. Let 0=i_0 < i_1 < ⋯ < i_s = d be integers. For any two nests of sets , with coranks i_0, ⋯, i_s, define the subvariety of (i_0,…,i_s;)

    (, ) = X_() ∩ X_(),

where X_() and X_() are defined in <ref>. Such a variety  is called a relative Richardson variety over S.


For any two permutations σ,τ, we also write (,)  as alternate notation for (,), where , are complete nests of sets with decreasing completions σ,τ, respectively. We write R_, for the open subscheme where all the defining rank conditions hold with equality, and use the notation R_σ,τ similarly.


We emphasize that we reserve the phrase “relative Richardson variety” for situations where S is smooth and irreducible and the flags , are versal, so that relative Richardson varieties share the geometric properties enjoyed by Richardson varieties, as summarized in Theorem <ref>.


Let S= k, let d=5, and 


    A^  =  {0,1,2,3,4}⊃{0,2,4}⊃∅,  
    B^ = {0,1,2,3,4}⊃{0,1,2}⊃∅ .


Then  is isomorphic to a Schubert variety with respect to the flag P^, parametrizing 2-dimensional subspaces V^3 with 

    V^3∩ P^2 ≥ 2   and   V^3∩ P^4 ≥ 1.



The cohomological statements in Theorem <ref> will be proved in Section <ref>. The rest of Theorem <ref> follows readily from the results of the previous section, as summarized below.

 
A relative Richardson variety  is normal and Cohen-Macaulay, of pure codimension (ωσ()) + (ωσ()) in (; i_0, ⋯, i_s), and the smooth locus of  is equal to the intersection of the smooth loci of the relative Schubert varieties X_A^ and X_B^.  The open subscheme R_, is dense in the smooth locus of .



Let σ = σ() and τ = σ(). Observe that = f^-1 ( ), where f: () →(i_0, ⋯, i_s; ) is the forgetful morphism from the complete flag variety. Since f is a fiber bundle with smooth irreducible fibers, we see that it suffices to prove the theorem for , i.e., for the case of complete flags. Note that

    = ( ; π^∗, π^∗),

where π: () → S is the structure map and  is the tautological bundle of ().  The flags ,π^∗, and  π^∗ are versal by Lemma <ref>, so Proposition <ref> and Corollary <ref> imply that  is normal and Cohen-Macaulay of pure codimension (ωσ)+(ωτ) in (), with smooth locus equal to the intersection of the smooth loci of (; π^∗) = X_σ() and D_τ(; π^∗) = X_τ(). 
Theorem <ref> and the description in Section <ref> of the singular locus of Schubert varieties show that the singular locus of  is a union of certain loci R_σ',τ' where σ' < σ or τ' < τ, and Proposition <ref> shows that each such locus has positive codimension. In particular, R_σ,τ is dense and contained in the smooth locus of .









 §.§ Cohomology of relative Richardson varieties
 

In order to understand the cohomology of the schemes , we will relate them to each other, and to the base S, via morphisms for which the total pushforward of the structure sheaf is trivial. Call a proper morphism of k-schemes π X→ Y -connected if _Y →π_* _X is an isomorphism, and call π a cohomological equivalence if it is -connected and  

    R^i π_* _X = 0   for all  i>0.
  
We note that the term “cohomological equivalence” is sometimes used in the literature specifically for birational morphisms (e.g. in <cit.>), but we use it in a more general way.
Note (see e.g. <cit.>) that if π is a cohomological equivalence then it induces canonical isomorphisms H^i(X,_X)≅ H^i (Y,_Y)
for all i≥ 0.  In particular, we have χ(X,_X) = χ(Y,_Y)
in this situation.  In what follows, we will make use of the following fact, which can be deduced from the Grothendieck spectral sequence: if f X' → X is a cohomological equivalence and π X → Y is any proper morphism, then π is a cohomological equivalence if and only if π∘ f is a cohomological equivalence. The purpose of this subsection is to prove the following theorem.

 
Let ^,^ be versal flags in a rank-d vector bundle  on a scheme S, and let , be any two nests of sets as defined in <ref>. Denote by R_, the resulting relative Richardson variety.
Let σ = σ() and τ = σ().

Then the image of the morphism → S is D_τ⋆σ^-1(^; ^), and the morphism → D_τ⋆σ^-1(^; ^) is a cohomological equivalence. 



Our strategy for proving Theorem <ref> is as follows. We first reduce to the case of complete flag varieties, i.e., varieties  (Corollary <ref>). We then show that the desired statement about  can be deduced from the statement for R_σ', τ', where σ',τ' is another pair of permutations with (σ') < (σ) and τ' ⋆σ'^-1 = τ⋆σ^-1 (Lemma <ref>). This reduces by induction to the case of the morphism R_id, τ⋆σ^-1→ S, which we show (Lemma <ref>).

Throughout the section, we fix the choice of S, , ^, ^. We do not assume in general that ^, ^ are versal, as several auxiliary results do not require this hypothesis; we will state it specifically when it is needed.
We begin with three useful criteria for cohomological equivalences.

 
If f: X → Y is a Grassmannian bundle, then f is a cohomological equivalence.



This follows from the fact that the structure sheaf of the Grassmannian variety (t,n) has no higher cohomology. This in turn follows from the Borel-Weil-Bott theorem, or can be seen more directly by induction on t, by observing that both the forgetful morphisms (t-1,t;n) →(t-1,n) and (t-1,t;n) →(t,n) are cohomological equivalences.


 
If f: X → Y is a birational morphism of normal, irreducible, projective varieties, then f is -connected.



See the proof of <cit.>.



Suppose X Y factors as X Z Y, where X→ Z is a closed immersion with ideal sheaf ℐ and Z→ Y is a ℙ^1-bundle. Then R^i π_* _X = 0 for all i>0.


  This argument may be found in <cit.>; we summarize it here for convenience. The statement holds for i>1 since all fibers of π have dimension at most 1.  For i=1, the exact sequence 0→ℐ→_Z → j_* _X → 0 yields the following portion of a long exact sequence.

    ⋯→ R^1π'_* _Z → R^1 π_*_X → R^2 π'_*ℐ→⋯

The first term is 0 since π' is a ℙ^1-bundle, and the third term is 0 since all fibers of π' have dimension 1. So the middle term is 0.







In what follows, we fix the following notation. If A,B ⊆ are finite sets, we write A < B to mean max A < min B. If , are two nests of sets, and j is an index with 0 < j < s, we denote the nested sets obtained by removing the jth set as follows.


    A_j^ = (A^i_0⊃⋯⊃A^i_j⊃⋯⊃ A^i_s),     
    B_j^ = (B^i_0⊃⋯⊃B^i_j⊃⋯⊃ B^i_s).


 
Let S be a smooth variety, with a rank d vector bundle  and two complete flags ,.
Suppose that , , j are as above, and that we have both 
    (A^i_j-1∖ A^i_j) >(A^i_j∖ A^i_j+1)  and 
    (B^i_j-1∖ B^i_j) > (B^i_j∖ B^i_j+1).

Then R_,→ R__j, _j is a cohomological equivalence, and it is a fiber bundle with smooth irreducible fibers.



For example, let =  = ({0,1,2,3}⊃{0,1}⊃∅), and j=1.  Then R_,→ R_A_j^, B_j^ is a Grassmannian 𝔾(1,^3) over a point. 


Observe that (σ(),σ()) does not contain (i_j, b) for any value of b (see Remark <ref>). So  and R_A_j^, B_j^ are defined by the same rank conditions, and thus  is the inverse image of R_A_j^, B_j^ under a forgetful morphism of partial flag varieties. The Lemma now follows from Fact <ref>.


Applying Lemma <ref> repeatedly gives the following.

 
Let ^, ^ be complete flags in a vector bundle  on a smooth scheme S. For any choice of ,,
the forgetful morphism

    R_σ(), σ()→ R_,

is a cohomological equivalence with smooth irreducible fibers.




Let ^, ^ be versal flags in a vector bundle  on a smooth scheme S. 
Given ,, define _j and _j as in Lemma <ref>. Suppose further that i_j+1 = i_j-1+2, and either
 
    (A^i_j-1∖ A^i_j) >(A^i_j∖ A^i_j+1)  or 
    (B^i_j-1∖ B^i_j) > (B^i_j∖ B^i_j+1).

Then the forgetful morphism π→ R__j, _j is a cohomological equivalence.



If both of these conditions hold, then the result follows from Lemma <ref>. Therefore we may assume without loss of generality that i_j+1 = i_j-1+2 and that 
 
    (A^i_j-1∖ A^i_j) >(A^i_j∖ A^i_j+1)  and 
    (B^i_j-1∖ B^i_j) < (B^i_j∖ B^i_j+1).

Writing {b} = B^i_j-1∖ B^i_j and {c} = B^i_j∖ B^i_j+1, we have b<c. Define C^ by “exchanging b and c,”  in other words C^i_j = C^i_j+1∪{b} and C^ is otherwise the same as . Notice that π factors as → R_,C^→, where the first morphism is a closed immersion, and the second is the structure map for a ^1-bundle,
by the argument in Lemma <ref>.  This implies that R^i π_* _ = 0 for i>0 (Fact <ref>). It remains to verify that π is -connected. The statement is local in R__j, _j, so we assume that R__j,_j is connected. Since it is also normal (Theorem <ref>), we may assume that R__j,_j is irreducible. By Fact <ref>, it suffices to check that π is birational. The open subscheme R_, is dense in  by Theorem <ref>, and likewise R__j,_j is dense in R__j,_j. At any point of R__j,_j, corresponding to flags V^, P^, Q^ in H, there is a unique choice of subspace V^i_j strictly between V^i_j-1 and V^i_j+1 which can be added to the flag V^ to produce a point of .  More precisely, over R_A^_j, B^_j, the morphism π is a structure morphism for a relative Schubert subvariety of a ^1-bundle consisting of all relative flags in this ^1-bundle  that coincide with a fixed flag.
This shows that in fact π restricts to an isomorphism R_, →R_A^_j, B^_j, which shows that π is indeed birational, which completes the proof.





Suppose σ,τ is a pair of permutations, and s is a simple transposition. Assume that ^, ^ are versal. Then R_σ⋆ s, τ→ S and R_σ, τ⋆ s→ S have the same image S', and if one is a cohomological equivalence then so is the other.



First, consider the case where σ⋆ s = σ. We may assume that τ⋆ s ≠τ. Let j be the index such that s transposes j and j-1. By Fact <ref>, σ_j-1 > σ_j, τ_j-1 < τ_j, and τ⋆ s = τ s. Let ,,C^ be the complete nests of sets associated to σ,τ, and τ s, respectively. 

Both R_σ,τ = and R_σ,τ s = R_,C^ have forgetful morphisms to R__j,_j; we record the relevant morphisms in the commuting diagram below.


Here i is a closed immersion, and f' is a ^1-bundle.  The maps f and f' are known to be cohomological equivalences, by Lemmas <ref> and <ref>. In particular, all three push-forwards π_∗_, π^j_∗_R__j, _j, π'_∗_R_, C^ agree. It follows that the scheme-theoretic image of all three has the same ideal sheaf on S (namely, the kernel of _S →π_∗_). Hence the image of all three morphisms π, π^j, π' is the same; call this image S'.

All three morphisms π, π^j, π' factor through S'. It follows from the Grothendieck spectral sequence that the morphism → S' is a cohomological equivalence if and only the morphism R__j,_j→ S' is a cohomological equivalence, if and only if the morphism R_, C^→ S' is a cohomological equivalence. This establishes the result in the case where σ⋆ s = σ.

Next, the case where τ⋆ s = τ follows from the first case by exchanging the flags. It remains to consider the case where τ⋆ s ≠τ and σ⋆ s ≠σ. In fact, this case follows from the first two: by Fact <ref> we have σ⋆ s = σ⋆ s ⋆ s = σ s and τ⋆ s = τ⋆ s ⋆ s = τ s, and the result follows by first relating R_σ⋆ s ⋆ s, τ to R_σ⋆ s, τ⋆ s and then relating R_σ⋆ s, τ⋆ s to R_σ, τ⋆ s ⋆ s.



 
Suppose that σ∈ S_d. For any scheme S with vector bundle  and complete flags ^, ^, the morphism R_id, σ(^, ^) → S is a closed immersion with image D_σ(^; ^).



Consider the functor of points of R_id,σ. A morphism T → R_id,σ corresponds to a morphism t: T → S and a complete flag ^ of t^∗ such that the permutation associated to (, t^∗^) is at most id and the permutation associated to (, t^∗^) is at most σ (where these statements are meant scheme-theoretically, i.e., as determinantal loci). The former condition is equivalent to saying that ^ is identical to t^∗^. So in fact morphisms T → R_id,σ correspond to morphisms t: T → S such that t^∗^, t^∗^ have associated permutation at least σ. But this is a description of the functor of D_σ(^; ^). We deduce that  the morphism R_id,σ→ S induces a bijection between morphisms T → R_σ, τ and morphisms T → D_σ(^; ^); the result follows.




We can now prove Theorems <ref> and <ref>.



Let σ = σ() and τ = σ(). By Corollary <ref> and the Grothendieck spectral sequence, → S and → S have the same image S', and one is a cohomological equivalence if and only if the other is. So it suffices to consider complete ,, i.e., the schemes . 

We prove the result by induction on (τ). The base case, τ = id, follows from Lemma <ref>. For the induction step, let s be a simple transposition such that (τ s) = (τ)-1. Then R_σ, τ = R_σ, (τ s) ⋆ s, so by Lemma <ref> the morphism R_σ⋆ s, τ s→ S has the same image S as → S, and is a cohomological equivalence if and only if → S is. By induction, this image is D_(τ s) ⋆ (σ⋆ s)^-1(; ) and both maps are cohomological equivalences. Finally, Fact <ref> implies that (τ s) ⋆ (σ⋆ s)^-1 = (τ s) ⋆ s^-1⋆σ^-1 = τ⋆σ^-1, so the image of both maps is D_τ⋆σ^-1(; ). This completes the induction.



Part (2) of Theorem <ref> is Theorem <ref>. Part (1) is Theorem <ref>.







§ BRILL-NOETHER VARIETIES AND RELATIVE RICHARDSON VARIETIES



This section describes an example of relative Richardson varieties arising in Brill-Noether theory, which is a crucial ingredient in <cit.>. Let E be an elliptic curve and L∈^d (E), and let V=H^0(E,L).  Suppose p and q are distinct closed points on E such that p-q is nontorsion in the Jacobian.  Assume d ≥ 1. Define two complete flags of V as follows.


    P^i = 
    V(-i p)    
    
    0    .
          
    Q^j = 
    V(-j q)    
    
    0    .





Let  be the rank d vector bundle on ^d E whose fiber over L is identified with H^0(E,L); more precisely  is the pushforward to ^d E of the Poincaré bundle on ^d E × E.
For discussion of Poincaré line bundles and this construction, see <cit.>. One must be careful in this construction to assume d ≥ 2g-1, so that  is a vector bundle, and to only define P^i for d-i ≥ 2g-1 for the same reason. In this case 2g-1 =1, so we have assumed d ≥ 1 and i < d.
Then the flags P^ and Q^, defined above for each L∈^d E, globalize to a pair of flags ^, ^.
 The flags ^ and ^ are versal.



Notice that , are transverse except when L≅𝒪(ap+bq) for some a,b> 0 and a+b=d. Since we assume p-q is non-torsion, the integers a,b are unique in this case, and we have


    P^i ∩ Q^j
    = max(d-i-j, 0)     (i,j) ≠ (a,b)
    
    1     (i,j) = (a,b).


It follows that if L≅𝒪(ap+bq) then the flags P^, Q^ are almost-transverse, i.e. their associated permutation is a simple transposition.
By Lemma <ref>, it is enough to show that the subscheme of ^d(E) over which  and  fail to be transverse consists of exactly these finitely many reduced points; the only issue to check here is reducedness.

Locally around a point L = _E( a p + b q), where a,b>0 with a+b=d, the scheme is question is defined by the condition P^a ∩ Q^b≥ 1 (where the scheme structure can be defined with a degeneracy condition for a map of vector bundles). This can be reformulated as the condition that P^a-1∩ Q^b and P^a∩ Q^b-1 are equal, viewing both of these as codimension 1 subspaces of P^a-1∩ Q^b-1. The assertion of reducedness amounts to showing that two sections of the ^1-bundle (^a-1∩^b-1) →^d E are transverse. This ^1-bundle can be identified with Sym^2 E, regarded as a ^1-bundle over ^2 E, by tensoring with ((a-1)p+(b-1)q).
The two sections are the effective divisors in Sym^2 E containing p and containing q, respectively. These loci meet transversely at p+q ∈Sym^2 E, since the tangent space there may be identified with T_p E × T_q E, and the tangent spaces to the curves identified with the tangent spaces of the two factors.


It follows that the twice-pointed Brill Noether varieties (E,p,q), studied in <cit.>, are examples of relative Richardson varieties.  See those papers for the definitions.

The schemes (E,p,q) are relative Richardson varieties over ^d(E).




 §.§ A conjecture in higher genus
 We conclude with a conjectural generalization to higher genus that would provide an intriguing generalization of some results in Brill-Noether theory. To state our conjecture, we generalize the notion of versality to partial flags in the natural way: ℓ partial flags on a scheme S are versal if the induced map from () to a product of partial flag varieties is smooth. One may then define relative Richardson varieties with respect to partial flags. Note that one must impose restrictions on nests of sets , that may be used to define such relative Richardson varieties: (σ()) must contain only elements (a,b) for which ^b is defined, and likewise for  and .

Let C be a curve of genus g with two marked points p,q. Fix an integer N ≥ 2g-1. For every point [L] ∈^N(C), the vector space H = H^0(C,L) has dimension N-g+1 and has two partial flags given by vanishing orders at p and q, namely

    P^a    =    H^0(C,L(-ap)) 
    
    Q^b    =    H^0(C,L(-bq))

The upper bounds on a,b ensure that P^a has codimension a in H, since L(-ap) and L(-bq) are nonspecial; this is analogous to the need in <cit.> to twist by a fixed divisor in order to work with lines bundles of degree at least 2g-1. This construction globalizes, giving a vector bundle  with partial flags ,, each with coranks 0,1,⋯,N-2g+1.

 
If (C,p,q) is a general twice-marked curve of genus g, then for all N≥ 2g-1 the flags , in  described above are versal on ^N(C).


This conjecture would show that for general curves C, Brill-Noether varieties G^r_d(C) are relative Richardson varieties (in the more general sense where partial flags are allowed). Indeed, by choosing N sufficiently large that there are integers a,b with N-a,N-b ≥ 2g-1 and N-a-b = d, one may embed G^r_d(C) in the Grassmannian bundle (r+1, ) by twisting by the divisor ap + bq. This gives the following isomorphism.

    G^r_d(C) ≅{ (L,V) ∈(r+1, ): V ⊆ (^a)_x   V ⊆ (^b)_x },

where  is the vector bundle on ^N(C) given by |_[L]≅ H^0(C,L(ap+bq)).
Conjecture <ref> would therefore imply that G^r_d(C) is isomorphic over ^d(C) to a relative Richardson variety.

More generally, Brill-Noether varieties with ramification G^r,α,β_d(C,p,q) (see <cit.> for definitions) may be described in a similar manner, and Conjecture <ref> also implies that they are isomorphic over ^d(C) to relative Richardson varieties.

We remark that Theorem <ref> generalizes readily to versal partial flags: one may either replace complete flag varieties with partial flag varieties throughout the proof, or deduce the general result from Theorem <ref> by locally extending the versal partial flags to versal complete flags. Conjecture <ref> would give a new way to study singularities of G^r,α,β_d(C,p,q) for general (C,p,q), and in particular would provide a new proof of the Gieseker-Petri Theorem and the main theorem of <cit.> characterizing the singular locus of G^r,α,β_d(C,p,q). It would also generalize these results from linear series to flags of linear series.















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