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Each connected graded, graded-commutative algebra $A$ of finite type over a field $\Bbbk$ of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the (higher) Koszul modules of $A$. In this note, we investigate the geometry of the support loci of these modules, called the resonance schemes of the algebra. When $A=\Bbbk\langle \Delta \rangle$ is the exterior Stanley-Reisner algebra associated to a finite simplicial complex $\Delta$, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group., Comment: 17 pages; accepted for publication in Journal of Algebraic Combinatorics

A fundamental problem at the confluence of algebraic geometry and representation theory is to describe the cohomology of line bundles on flag varieties over a field of characteristic p. When p=0, the solution is given by the celebrated Borel-Weil-Bott Theorem, while for p>0 the problem is widely open. In this note we describe a collection of open questions that arise from the study of particular cases of the general theory, focusing on their combinatorial and commutative algebra aspects.

We prove an effective stabilization result for the sheaf cohomology groups of line bundles on flag varieties parametrizing complete flags in k^n, as well as for the sheaf cohomology groups of polynomial functors applied to the cotangent sheaf Omega on projective space. In characteristic zero, these are natural consequences of the Borel-Weil-Bott theorem, but in characteristic p>0 they are non-trivial. Unlike many important contexts in modular representation theory, where the prime characteristic p is assumed to be large relative to n, in our study we fix p and let n go to infinity. We illustrate the general theory by providing explicit stable cohomology calculations in a number of cases of interest. Our examples yield cohomology groups where the number of indecomposable summands has super-polynomial growth, and also show that the cohomological degrees where non-vanishing occurs do not form a connected interval. In the case of polynomial functors of Omega, we prove a Kunneth formula for stable cohomology, and show the invariance of stable cohomology under Frobenius, which combined with the Steinberg tensor product theorem yields calculations of stable cohomology for an interesting class of simple polynomial functors arising in the work of Doty. The results in the special case of symmetric powers of Omega provide a nice application to commutative algebra, yielding a sharp vanishing result for Koszul modules of finite length in all characteristics.

The resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial, and geometric contexts. We discuss their scheme structure in a general algebraic setting and introduce various properties that ensure the reducedness of the associated projective resonance scheme. We prove an asymptotic formula for the Hilbert series of the associated Koszul module, then discuss applications to vector bundles on algebraic curves and to Chen ranks formulas for finitely generated groups, with special emphasis on K\"ahler and right-angled Artin groups., Comment: 36 pages. Final version, to appear in the Journal fuer die reine und angewandte Mathematik (Crelle)

For a finite dimensional vector space V of dimension n, we consider the incidence correspondence (or partial flag variety) X in P(V) x P(V*), parametrizing pairs consisting of a point and a hyperplane containing it. We completely characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on X in characteristic p>0. If n=3 then X is the full flag variety of V, and the characterization is contained in the thesis of Griffith from the 70s. In characteristic 0, the cohomology groups are described for all V by the Borel-Weil-Bott theorem. Our strategy is to recast the problem in terms of computing cohomology of (twists of) divided powers of the cotangent sheaf on projective space, which we then study using natural truncations induced by Frobenius, along with careful estimates of Castelnuovo-Mumford regularity. When n=3, we recover the recursive description of characters from recent work of Linyuan Liu, while for general n we give character formulas for the cohomology of a restricted collection of line bundles. Our results suggest truncated Schur functions as the natural building blocks for the cohomology characters.

We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace in the second wedge product of a vector space. Previously Koszul modules of finite length have been used to give a proof of Green's Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves on K3 surfaces and to skew-symmetric degeneracy loci. We also show that the stability of sufficiently positive rank 2 vector bundles on curves is governed by resonance., Comment: 28 pages. To appear in Selecta Math

We compute the rational Borel-Moore homology groups for affine determinantal varieties in the spaces of general, symmetric, and skew-symmetric matrices, solving a problem suggested by the work of Pragacz and Ratajski. The main ingredient is the relation with Hartshorne's algebraic de Rham homology theory, and the calculation of the singular cohomology of matrix orbits, using the methods of Cartan and Borel. We also establish the degeneration of the \v{C}ech-de Rham spectral sequence for determinantal varieties, and compute explicitly the dimensions of de Rham cohomology groups of local cohomology with determinantal support, which are analogues of Lyubeznik numbers first introduced by Switala. Additionally, in the case of general matrices we further determine the Hodge numbers of the singular cohomology of matrix orbits and of the Borel-Moore homology of their closures, based on Saito's theory of mixed Hodge modules., Comment: 28 pages, v2: added results on mixed Hodge structures. New sections: 2.2, 4.3, 7

The study of Chow varieties of decomposable forms lies at the confluence of algebraic geometry, commutative algebra, representation theory and combinatorics. There are many open questions about homological properties of Chow varieties and interesting classes of modules supported on them. The goal of this note is to survey some fundamental constructions and properties of these objects, and to propose some new directions of research. Our main focus will be on the study of certain maximal Cohen-Macaulay modules of covariants supported on Chow varieties, and on defining equations and syzygies. We also explain how to assemble Tor groups over Veronese subalgebras into modules over a Chow variety, leading to a result on the polynomial growth of these groups., Comment: 20 pages; Dedicated to Bernd Sturmfels on the occasion of his 60th birthday

Hermite reciprocity refers to a series of natural isomorphisms involving compositions of symmetric, exterior, and divided powers of the standard $SL_2$-representation. We survey several equivalent constructions of these isomorphisms, as well as their recent applications to Green's Conjecture on syzygies of canonical curves. The most geometric approach to Hermite reciprocity is based on an idea of Voisin to realize certain multilinear constructions cohomologically by working on a Hilbert scheme of points. We explain how in the case of ${\bf P}^1$ this can be reformulated in terms of cohomological properties of Schwarzenberger bundles. We then proceed to study these bundles from several perspectives: We show that their exterior powers have supernatural cohomology, arising as special cases of a construction of Eisenbud and Schreyer. We recover basic properties of secant varieties $\Sigma$ of rational normal curves (normality, Cohen-Macaulayness, rational singularities) by considering their desingularizations via Schwarzenberger bundles, and applying the Kempf-Weyman geometric technique. We show that Hermite reciprocity is equivalent to the self-duality of the unique rank one Ulrich module on the affine cone $\widehat{\Sigma}$ of some secant variety, and we explain how for a Schwarzenberger bundle of rank $k$ and degree $d\ge k$, Hermite reciprocity can be viewed as the unique (up to scaling) non-zero section of $(Sym^k\mathcal{E})(-d+k-1)$., Comment: 26 pages

We prove a case of a positivity conjecture of Mihalcea-Singh, concerned with the local Euler obstructions associated to the Schubert stratification of the Lagrangian Grassmannian LG(n,2n). Combined with work of Aluffi-Mihalcea-Sch\"urmann-Su, this further implies the positivity of the Mather classes for Schubert varieties in LG(n,2n), which Mihalcea-Singh had verified for the other cominuscule spaces of classical Lie type. Building on the work of Boe and Fu, we give a positive recursion for the local Euler obstructions, and use it to show that they provide a positive count of admissible labelings of certain trees, analogous to the ones describing Kazhdan-Lusztig polynomials. Unlike in the case of the Grassmannians in types A and D, for LG(n,2n) the Euler obstructions e_{y,w} may vanish for certain pairs (y,w) with y <= w in the Bruhat order. Our combinatorial description allows us to classify all the pairs (y,w) for which e_{y,w}=0. Restricting to the big opposite cell in LG(n,2n), which is naturally identified with the space of n x n symmetric matrices, we recover the formulas for the local Euler obstructions associated with the matrix rank stratification.

The goal of this note is to explain a derivation of the formulas for the local Euler obstructions of determinantal varieties of general, symmetric and skew-symmetric matrices, by studying the invariant de Rham complex and using character formulas for simple equivariant $D$-modules. These calculations are then combined with standard arguments involving Kashiwara's local index formula and the description of characteristic cycles of simple equivariant $D$-modules. The formulas are implicit in the work of Boe and Fu, and in the case of general matrices they have also been obtained recently by Gaffney--Grulha--Ruas, for skew-symmetric matrices by Promtapan and Rim\'anyi, and for all cases by Zhang., Comment: 19 pages. Final version

Let $R=\Bbbk[x_1,\dots,x_n]$ be a polynomial ring over a field $\Bbbk$ and let $I\subset R$ be a monomial ideal preserved by the natural action of the symmetric group $\mathfrak S_n$ on $R$. We give a combinatorial method to determine the $\mathfrak S_n$-module structure of $\mathrm{Tor}_i(I,\Bbbk)$. Our formula shows that $\mathrm{Tor}_i(I,\Bbbk)$ is built from induced representations of tensor products of Specht modules associated to hook partitions, and their multiplicities are determined by topological Betti numbers of certain simplicial complexes. This result can be viewed as an $\mathfrak S_n$-equivariant analogue of Hochster's formula for Betti numbers of monomial ideals. We apply our results to determine extremal Betti numbers of $\mathfrak S_n$-invariant monomial ideals, and in particular recover formulas for their Castelnuovo--Mumford regularity and projective dimension. We also give a concrete recipe for how the Betti numbers change as we increase the number of variables, and in characteristic zero (or $>n$) we compute the $\mathfrak S_n$-invariant part of $\mathrm{Tor}_i(I,\Bbbk)$ in terms of $\mathrm{Tor}$ groups of the unsymmetrization of $I$., Comment: 31 pages

We determine explicitly the Hodge ideals for the determinant hypersurface as an intersection of symbolic powers of determinantal ideals. We prove our results by studying the Hodge and weight filtrations on the mixed Hodge module O_X(*Z) of regular functions on the space X of n x n matrices, with poles along the divisor Z of singular matrices. The composition factors for the weight filtration on O_X(*Z) are pure Hodge modules with underlying D-modules given by the simple GL-equivariant D-modules on X, where GL is the natural group of symmetries, acting by row and column operations on the matrix entries. By taking advantage of the GL-equivariance and the Cohen-Macaulay property of their associated graded, we describe explicitly the possible Hodge filtrations on a simple GL-equivariant D-module, which are unique up to a shift determined by the corresponding weights. For non-square matrices, O_X(*Z) is naturally replaced by the local cohomology modules H^j_Z(X,O_X), which turn out to be pure Hodge modules. By working out explicitly the Decomposition Theorem for some natural resolutions of singularities of determinantal varieties, and using the results on square matrices, we determine the weights and the Hodge filtration for these local cohomology modules., Comment: 16 pages, v2: updated discussion of generation level, and added Remark 4.2 on alternate approach to calculation of weight filtration

We prove the case t = 2 of a conjecture of Bruns-Conca-Varbaro, describing the minimal relations between the t x t minors of a generic matrix. Interpreting these relations as polynomial functors, and applying transpose duality as in the work of Sam-Snowden, this problem is equivalent to understanding the relations satisfied by t x t generalized permanents. Our proof follows by combining Koszul homology calculations on the minors side, with a study of subspace varieties on the permanents side, and with the Kempf-Weyman technique (on both sides)., Comment: 21 pages. To appear in Advances in Mathematics

We extend the theory of Koszul modules to the bi-graded case, and prove a vanishing theorem that allows us to show that the Canonical Ribbon Conjecture of Bayer and Eisenbud holds over a field of characteristic zero or at least equal to the Clifford index. Our results confirm a conjecture of Eisenbud and Schreyer regarding the characteristics where the generic statement of Green's conjecture holds. They also recover and extend to positive characteristics results due to Aprodu and Voisin asserting that Green's Conjecture holds for generic curves of each gonality., Comment: 23 pages

For a polynomial ring S in n variables, we consider the natural action of the symmetric group S_n on S by permuting the variables. For an S_n-invariant monomial ideal I in S and j >= 0, we give an explicit recipe for computing the modules Ext^j(S/I,S), and use this to describe the projective dimension and regularity of I. We classify the S_n-invariant monomial ideals that have a linear free resolution, and also characterize those which are Cohen-Macaulay. We then consider two settings for analyzing the asymptotic behavior of regularity: one where we look at powers of a fixed ideal I, and another where we vary the dimension of the ambient polynomial ring and examine the invariant monomial ideals induced by I. In the first case we determine the asymptotic regularity for those ideals I that are generated by the S_n-orbit of a single monomial by solving an integer linear optimization problem. In the second case we describe the behavior of regularity for any I, recovering a recent result of Murai.

For fixed weights w_1,...,w_n, and for d>0, we let B denote a collection of d*n balls, with d balls of weight w_i for each i=1,...,n. We consider the problem of assigning the balls to n bins with capacities C_1,...,C_n, in such a way that each bin is assigned d balls, without exceeding its capacity. When d>>0, we give sufficient criteria for the feasibility of this problem, which coincide up to explicit constants with the natural set of necessary conditions. Furthermore, we show that our constants are optimal when the weights w_i are distinct. The feasibility criteria that we present here are used elsewhere (in commutative algebra) to study the asymptotic behavior of the Castelnuovo-Mumford regularity of symmetric monomial ideals.

We describe a Macaulay2 package for computing Schur complexes. This package expands on the ChainComplexOperations package by David Eisenbud., Comment: 9 pages, v2: revised Section 2.1. To appear in the Journal of Software for Algebra and Geometry. Updated package, available at: https://github.com/mperlm3/SchurComplexes

We prove a strong vanishing result for finite length Koszul modules, and use it to derive Green's conjecture for every g-cuspidal rational curve over an algebraically closed field k with char(k) = 0 or char(k) >= (g+2)/2. As a consequence, we deduce that the general canonical curve of genus g satisfies Green's conjecture in this range. Our results are new in positive characteristic, whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by Voisin. Our strategy involves establishing two key results of independent interest: (1) we describe an explicit, characteristic-independent version of Hermite reciprocity for sl_2-representations; (2) we completely characterize, in arbitrary characteristics, the (non-)vanishing behavior of the syzygies of the tangential variety to a rational normal curve., Comment: minor edits, 42 pages, to appear in Invent. Math

We let S denote the ring of polynomial functions on the space of m x n matrices, and consider the action of the group GL = GL_m x GL_n via row and column operations on the matrix entries. For a GL-invariant ideal I in S we show that the linear strands of its minimal free resolution translate via the BGG correspondence to modules over the general linear Lie superalgebra gl(m|n). When I is the ideal generated by the GL-orbit of a highest weight vector, we give a conjectural description of the classes of these gl(m|n)-modules in the Grothendieck group, and prove that our prediction is correct for the first strand of the minimal free resolution., Comment: To appear in Acta Mathematica Vietnamica, special issue: The Prospects for Commutative Algebra

We provide a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace K inside the second exterior product of a vector space, as well as a sharp upper bound for its Hilbert function. This purely algebraic statement has interesting applications to the study of a number of invariants associated to finitely generated groups, such as the Alexander invariants, the Chen ranks, or the degree of growth and nilpotency class. For instance, we explicitly bound the aforementioned invariants in terms of the first Betti number for the maximal metabelian quotients of (1) the Torelli group associated to the moduli space of curves; (2) nilpotent fundamental groups of compact Kaehler manifolds; (3) the Torelli group of a free group., Comment: 25 pages. Final version, to appear in Duke Math. Journal

We give an explicit recipe for determining iterated local cohomology groups with support in ideals of minors of a generic matrix in characteristic zero, expressing them as direct sums of indecomposable D-modules. For non-square matrices these indecomposables are simple, but this is no longer true for square matrices where the relevant indecomposables arise from the pole order filtration associated with the determinant hypersurface. Specializing our results to a single iteration, we determine the Lyubeznik numbers for all generic determinantal rings, thus answering a question of Hochster.

The study of homological invariants such as Tor, Ext and local cohomology modules constitutes an important direction in commutative algebra. Explicit descriptions of these invariants are notoriously difficult to find and often involve combining an array of techniques from other fields of mathematics. In recent years tools from algebraic geometry and representation theory have been successfully employed in order to shed some light on the structure of homological invariants associated with determinantal rings. The goal of this notes is to survey some of these results, focusing on examples in an attempt to clarify some of the more technical statements., Comment: appeared in honorary volume dedicated to Prof. Dorin Popescu

We consider the space X = Sym^3(C^2) of binary cubic forms, equipped with the natural action of the group GL_2 of invertible linear transformations of C^2. We describe explicitly the category of GL_2-equivariant coherent D_X-modules as the category of representations of a quiver with relations. We show moreover that this quiver is of tame representation type and we classify its indecomposable representations. We also give a construction of the simple equivariant D_X-modules (of which there are 14), and give formulas for the characters of their underlying GL_2-representations. We conclude the article with an explicit calculation of (iterated) local cohomology groups with supports given by orbit closures.

We analyze the vanishing and non-vanishing behavior of the graded Betti numbers for Segre embeddings of products of projective spaces. We give lower bounds for when each of the rows of the Betti table becomes non-zero, and prove that our bounds are tight for Segre embeddings of products of P^1. This generalizes results of Rubei concerning the Green-Lazarsfeld property N_p for Segre embeddings. Our methods combine the Kempf-Weyman geometric technique for computing syzygies, the Ein-Erman-Lazarsfeld approach to proving non-vanishing of Betti numbers, and the theory of algebras with straightening laws.

We consider the ring S=C[x_ij] of polynomial functions on the vector space C^(m x n) of complex m x n matrices. We let GL= GL_m x GL_n and consider its action via row and column operations on C^(m x n) (and the induced action on S). For every GL-invariant ideal I in S and every j>=0, we describe the decomposition of the modules Ext^j_S(S/I,S) into irreducible GL-representations. For any inclusion I into J of GL-invariant ideals we determine the kernels and cokernels of the induced maps Ext^j_S(S/I,S) -> Ext^j_S(S/J,S). As a consequence of our work, we give a formula for the regularity of the powers and symbolic powers of generic determinantal ideals, and in particular we determine which powers have a linear minimal free resolution. As another consequence, we characterize the GL-invariant ideals I in S for which the induced maps Ext^j_S(S/I,S) -> H_I^j(S) are injective. In a different direction we verify that Kodaira vanishing, as described in work of Bhatt-Blickle-Lyubeznik-Singh-Zhang, holds for determinantal thickenings., Comment: minor changes, to appear in Proc. Lond. Math. Soc

We determine the Bernstein-Sato polynomials for the ideal of maximal minors of a generic m x n matrix, as well as for that of sub-maximal Pfaffians of a generic skew-symmetric matrix of odd size. As a corollary, we obtain that the Strong Monodromy Conjecture holds in these two cases.

We compute the local cohomology modules H_Y^(X,O_X) in the case when X is the complex vector space of n x n symmetric, respectively skew-symmetric matrices, and Y is the closure of the GL-orbit consisting of matrices of any fixed rank, for the natural action of the general linear group GL on X. We describe the D-module composition factors of the local cohomology modules, and compute their multiplicities explicitly in terms of generalized binomial coefficients. One consequence of our work is a formula for the cohomological dimension of ideals of even minors of a generic symmetric matrix: in the case of odd minors, this was obtained by Barile in the 90s. Another consequence of our work is that we obtain a description of the decomposition into irreducible GL-representations of the local cohomology modules (the analogous problem in the case when X is the vector space of m x n matrices was treated in earlier work of the authors).

We compute the characters of the simple GL-equivariant holonomic D-modules on the vector spaces of general, symmetric and skew-symmetric matrices. We realize some of these D-modules explicitly as subquotients in the pole order filtration associated to the determinant/Pfaffian of a generic matrix, and others as local cohomology modules. We give a direct proof of a conjecture of Levasseur in the case of general and skew-symmetric matrices, and provide counterexamples in the case of symmetric matrices. The character calculations are used in subsequent work with Weyman to describe the D-module composition factors of local cohomology modules with determinantal and Pfaffian support., Comment: Reorganized the material according to the referee's suggestions. To appear in Compos. Math

For d > 1, we consider the Veronese map of degree d on a complex vector space W , Ver_d : W -> Sym^d W , w -> w^d , and denote its image by Z. We describe the characters of the simple GL(W)-equivariant holonomic D-modules supported on Z. In the case when d is 2, we obtain a counterexample to a conjecture of Levasseur by exhibiting a GL(W)-equivariant D-module on the Capelli type representation Sym^2 W which contains no SL(W)-invariant sections. We also study the local cohomology modules H_Z^j(S), where S is the ring of polynomial functions on the vector space Sym^d W. We recover a result of Ogus showing that there is only one local cohomology module that is non-zero (namely in degree j = codim(Z)), and moreover we prove that it is a simple D-module and determine its character., Comment: minor changes, to appear in Trans. AMS