arXiv subjects

Michael Perlman

Publications and source records attributed to Michael Perlman.

14 recordsLinked to original sources

arXiv2026

Microlocal Bernstein--Sato polynomials on singular ambient varieties

We introduce the microlocal Bernstein--Sato polynomial of a function on a possibly singular ambient variety, extending the theory of Saito. We show that, contrary to the smooth ambient setting, these polynomials are not generally equal to the reduced $b$-functions obtained by removing the trivial root. We define the minimal exponent and use it to study the singularities of the divisor and the Hodge filtration on local cohomology. Our main results include a generalization of Saito's theorem relating the minimal exponent to rational singularities, a characterization of purity of local cohomology, a Thom--Sebastiani formula for the minimal exponent, and a linear combination formula for Bernstein--Sato polynomials of ideals. When the ambient variety is a complete intersection with rational singularities, we provide effective algorithms for these Bernstein--Sato polynomials and implement them in Macaulay2.

math.AG↗

arXiv2024

Local cohomology with support in Schubert varieties

This paper is concerned with local cohomology sheaves on generalized flag varieties supported in closed Schubert varieties, which carry natural structures as (mixed Hodge) D-modules. We employ Kazhdan--Lusztig theory and Saito's theory of mixed Hodge modules to describe a general strategy to calculate the simple composition factors, Hodge filtration, and weight filtration on these modules. Our main tool is the Grothendieck--Cousin complex, introduced by Kempf, which allows us to relate the local cohomology modules in question to parabolic Verma modules over the corresponding Lie algebra. We show that this complex underlies a complex of mixed Hodge modules, and is thus endowed with Hodge and weight filtrations. As a consequence, strictness implies that computing cohomology commutes with taking associated graded with respect to both of these filtrations. We execute this strategy to calculate the composition factors and weight filtration for Schubert varieties in the Grassmannian, in particular showing that the weight filtration is controlled by the augmented Dyck patterns of Raicu--Weyman. As an application, upon restriction to the opposite big cell, we recover the simple composition factors and weight filtration on local cohomology with support in generic determinantal varieties.

math.AG↗

arXiv2024

Ideals preserved by linear changes of coordinates in positive characteristic

We consider the polynomial ring in finitely many variables over an algebraically closed field of positive characteristic, and initiate the systematic study of ideals preserved by the action of the general linear group by changes of coordinates. We show that these ideals are classified by sets of carry patterns, which are finite sequences of integers introduced by Doty in the study of representation theory of the polynomial ring. We provide an algorithm to decompose an invariant ideal as a sum of carry ideals with no redundancies. Next, we study the conditions under which one carry ideal is contained in another, and completely characterize the image of the multiplication map between the space of linear forms and a subrepresentation of forms of degree d. Finally, we begin an investigation into free resolutions of these ideals. Our results are most explicit in the case of carry ideals in two variables, where we completely describe the monomial generators and syzygies using base-p expansions of the parameters involved, and we provide a formula for the structure of the Tor modules in the Grothendieck group of representations.

math.AC↗

arXiv2023

Equivariant D-modules on 2x2xn hypermatrices

We study D-modules and related invariants on the space of 2 x 2 x n hypermatrices for n >= 3, which has finitely many orbits under the action of G = GL_2 x GL_2 x GL_n. We describe the category of coherent G-equivariant D-modules as the category of representations of a quiver with relations. We classify the simple equivariant D-modules, determine their characteristic cycles and find special representations that appear in their G-structures. We determine the explicit D-module structure of the local cohomology groups with supports given by orbit closures. As a consequence, we calculate the Lyubeznik numbers and intersection cohomology groups of the orbit closures. All but one of the orbit closures have rational singularities: we use local cohomology to prove that the one exception is neither normal nor Cohen--Macaulay. While our results display special behavior in the cases n=3 and n=4, they are completely uniform for n >= 5.

math.AG↗

arXiv2022

Socle degrees for local cohomology modules of thickenings of maximal minors and sub-maximal Pfaffians

Let $S$ be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let $I$ be the determinantal ideal of maximal minors or $\operatorname{Pf}$ the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu--Weyman--Witt to determine the $S$-module structures of $\operatorname{Ext}^j_S(S/I^t, S)$ and $\operatorname{Ext}^j_S(S/\operatorname{Pf}^t, S)$, from which we get the degrees of generators of these $\operatorname{Ext}$ modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang on the socle degrees of local cohomology modules of the form $H^j_\mathfrak{m}(S/I^t)$.

math.AC↗

arXiv2022

Equivariant resolutions over Veronese rings

Working in a polynomial ring $S=\mathbf{k}[x_1,\ldots,x_n]$ where $\mathbf{k}$ is an arbitrary commutative ring with $1$, we consider the $d^{th}$ Veronese subalgebras $R=S^{(d)}$, as well as natural $R$-submodules $M=S^{(\geq r, d)}$ inside $S$. We develop and use characteristic-free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple $GL_n(\mathbf{k})$-equivariant minimal free $R$-resolutions for the quotient ring $\mathbf{k}=R/R_+$ and for these modules $M$. These also lead to elegant descriptions of $\mathrm{Tor}^R_i(M,M')$ for all $i$ and $\mathrm{Hom}_R(M,M')$ for any pair of these modules $M,M'$.

math.AC↗

arXiv2021

Mixed Hodge structure on local cohomology with support in determinantal varieties

We employ the inductive structure of determinantal varieties to calculate the mixed Hodge module structure of local cohomology modules with determinantal support. We show that the weight of a simple composition factor is uniquely determined by its support and cohomological degree. As a consequence, we obtain the equivariant structure of the Hodge filtration on each local cohomology module. Finally, as an application, we provide a formula for the generation level of the Hodge filtration on these modules.

math.AG↗

arXiv2020

Hodge ideals for the determinant hypersurface

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.

math.AG↗

arXiv2019

Relations between the 2x2 minors of a generic matrix

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).

math.AC↗

arXiv2018

Computing Schur complexes

We describe a Macaulay2 package for computing Schur complexes. This package expands on the ChainComplexOperations package by David Eisenbud.

math.AC↗

arXiv2018

Equivariant D-modules on alternating senary 3-tensors

Let X be the third exterior power of a six-dimensional complex vector space, equipped with the natural action of the group GL_6(C) of invertible linear transformations of C^6. We describe explicitly the category of GL_6(C)-equivariant coherent D_X-modules as the category of representations of a quiver with relations, which has finite representation type. We give a construction of the six simple equivariant D_X-modules and give formulas for the characters of their underlying GL_6(C)-structures. We describe the (iterated) local cohomology groups with supports given by orbit closures, determining, in particular, the Lyubeznik numbers associated to the orbit closures.

math.AC↗

arXiv2018

Lyubeznik numbers for Pfaffian rings

We study the structure of local cohomology with support in Pfaffian varieties as a module over the Weyl algebra D_X of differential operators on the space X of n x n complex skew-symmetric matrices. The simple composition factors of these modules are known by the work of Raicu-Weyman in 2016, and when n is odd, the general theory implies that the local cohomology modules are semi-simple. When n is even, we show that the local cohomology is a direct sum of indecomposable modules coming from the pole order filtration of the Pfaffian hypersurface. We then determine the Lyubeznik numbers for Pfaffian rings by computing local cohomology with support in the homogeneous maximal ideal of the indecomposable summands referred to above.

math.AC↗

arXiv2018

Equivariant D-modules on 2x2x2 hypermatrices

Let V be the space of 2x2x2 complex hypermatrices, endowed with the natural group action of GL=GL(2,C)^3. The category of GL-equivariant coherent D-modules on V is equivalent to the category of representations of a quiver with relations. In this article, we give a construction of each simple object and study their GL-equivariant structure. Using this information, we go on to explicitly describe the corresponding quiver with relations. As an application, we compute all iterations of local cohomology with support in the orbit closures of V.

math.AC↗

arXiv2017

Regularity and cohomology of Pfaffian thickenings

Let $S$ be the coordinate ring of the space of $n\times n$ complex skew-symmetric matrices. This ring has an action of the group $\textrm{GL}_n(\mathbb{C})$ induced by the action on the space of matrices. For every invariant ideal $I\subseteq S$, we provide an explicit description of the modules $\textrm{Ext}^{\bullet}_S(S/I,S)$ in terms of irreducible representations. This allows us to give formulas for the regularity of basic invariant ideals and (symbolic) powers of ideals of Pfaffians, as well as to characterize when these ideals have a linear free resolution. In addition, given an inclusion of invariant ideals $I\supseteq J$, we compute the (co)kernel of the induced map $\textrm{Ext}^j_S(S/I,S)\to \textrm{Ext}^j_S(S/J,S)$ for all $j\geq 0$. As a consequence, we show that if an invariant ideal $I$ is unmixed, then the induced maps $\textrm{Ext}_S^j(S/I,S)\to H_I^j(S)$ are injective, answering a question of Eisenbud-Musta\c{t}\u{a}-Stillman in the case of Pfaffian thickenings. Finally, using our Ext computations and local duality, we verify an instance of Kodaira vanishing in the sense described in the recent work of Bhatt-Blickle-Lyubeznik-Singh-Zhang.

math.AC↗