Patricia Klein: Research

Papers and preprints:

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† denotes undergraduate mentee.

Some algebraic properties of ASM varieties (with Ilani Axelrod-Freed†, Hanson Hao†, Matthew Kendall†, and Yuyuan Luo†)

Submitted, 23 pages.
arXiv:2505.10480

Fulton’s matrix Schubert varieties are affine varieties that arise in the study of Schubert calculus in the complete flag variety. Weigandt showed that arbitrary intersections of matrix Schubert varieties, now called ASM varieties, are indexed by alternating sign matrices (ASMs), objects with a long history in enumerative combinatorics. It is very difficult to assess Cohen–Macaulayness of ASM varieties or to compute their codimension, though these properties are well understood for matrix Schubert varieties due to work Fulton. In this paper we study these properties of ASM varieties with a focus on the relationship between a pair of ASMs and their direct sum. We also consider ASM pattern avoidance from an algebro-geometric perspective.
Algebra and geometry of ASM weak order (with Laura Escobar and Anna Weigandt)

Submitted, 33 pages.
arXiv:2502.19266

Much of modern Schubert calculus is centered on Schubert varieties in the complete flag variety \(\mathcal{F}\ell(n)\) and on their classes in the integral cohomology ring \(\mathcal{H}^\ast(\mathcal{F}\ell(n))\). Under the Borel isomorphism, these classes are represented by distinguished polynomials called Schubert polynomials, introduced by Lascoux and Schützenberger. Schubert polynomials form an additive basis of \(\mathcal{H}^\ast(\mathcal{F}\ell(n))\).

Knutson and Miller showed that Schubert polynomials are multidegrees of matrix Schubert varieties, affine varieties introduced by Fulton, which are closely related to Schubert varieties. Many roads to studying Schubert polynomials pass through unions and intersections of matrix Schubert varieties. The third author showed that the natural indexing objects of arbitrary intersections of matrix Schubert varieties are alternating sign matrices (ASMs), which had previously enjoyed rich study in enumerative combinatorics and, as the six–vertex ice model, in statistical mechanics. Intersections of matrix Schubert varieties are now called ASM varieties. Every ASM variety is expressible as a union of matrix Schubert varieties.

Many fundamental algebro-geometric invariants (e.g., codimension, degree, and Castelnuovo-Mumford regularity) are well understood combinatorially for matrix Schubert varieties, substantially via the combinatorics of strong Bruhat order on Sn. The extension of strong order to ASM(\(n\)), the set of \(n \times n\) ASMs, has so far not borne as much algebro-geometric fruit for ASM varieties.

Hamaker and Reiner proposed an extension of weak Bruhat order from \(S_n\) to ASM(\(n\)), which they studied from a combinatorial perspective. In the present paper, we place this work on algebro-geometric footing. We use weak order on ASMs to give a characterization of codimension of ASM varieties. We also show that weak order operators commute with K-theoretic divided difference operators and that they satisfy the same derivative formula that facilitated the first general combinatorial computation of Castelnuovo-Mumford regularity of matrix Schubert varieties. Finally, we build from these results to generalizations that apply to arbitrary unions of matrix Schubert varieties.
Polarization and Gorenstein liaison (with Sara Faridi, Jenna Rajchgot, and Alexandra Seceleanu)

Submitted, 30 pages.
arXiv:2406.19985

A major open question in the theory of Gorenstein liaison is whether or not every arithmetically Cohen–Macaulay subscheme of \(\mathbb{P}^n\) can be G-linked to a complete intersection. Migliore and Nagel showed that, if such a scheme is generically Gorenstein (e.g., reduced), then, after re-embedding so that it is viewed as a subscheme of \(\mathbb{P}^{n+1}\), indeed it can be G-linked to a complete intersection. Motivated by this result, we consider techniques for constructing G-links on a scheme from G-links on a closely related reduced scheme.

Polarization is a tool for producing a squarefree monomial ideal from an arbitrary monomial ideal. Basic double G-links on squarefree monomial ideals can be induced from vertex decompositions of their Stanley–Reisner complexes. Given a monomial ideal \(I\) and a vertex decomposition of the Stanley–Reisner complex of its polarization \(\mathcal{P}(I)\), we give conditions that allow for the lifting of an associated basic double G-link of \(\mathcal{P}(I)\) to a basic double G-link of \(I\) itself. We use the relationship we develop in the process to show that the Stanley–Reisner complexes of polarizations of artinian monomial ideals and of stable Cohen–Macaulay monomial ideals are vertex decomposable, recovering and strengthening the recent result of Floystad and Mafi that these complexes are shellable.

We then introduce and study polarization of a Gröbner basis of an arbitrary homogeneous ideal and give a relationship between geometric vertex decomposition of a polarization and elementary G-biliaison that is analogous to our result on vertex decomposition and basic double G-linkage.
Simplicial complexes and matroids with vanishing \(T^2\) (with Alexandru Constantinescu, Thai Thanh Nguyen, Anurag Singh, and Lorenzo Venturello )

Electronic Journal of Combinatorics 32 (2025), no. 2, Paper no. 2.12.
Journal version
arXiv:2406.02440

We investigate quotients by radical monomial ideals for which \(T^2\), the second cotangent cohomology module, vanishes. The dimension of the graded components of \(T^2\), and thus their vanishing, depends only on the combinatorics of the corresponding simplicial complex. We give both a complete characterization and a full list of one dimensional complexes with \(T^2=0\). We characterize the graded components of \(T^2\) when the simplicial complex is a uniform matroid. Finally, we show that \(T^2\) vanishes for all matroids of corank at most two and conjecture that all connected matroids with vanishing \(T^2\) are of corank at most two.
On basic double G-links of squarefree monomial ideals (with Matthew Koban† and Jenna Rajchgot)

Journal of Commutative Algebra, 16 (2024), No. 2, 213-229.
Journal version
arXiv:2209.00119

Nagel and Römer introduced the class of weakly vertex decomposable simplicial complexes, which include matroid, shifted, and Gorenstein complexes as well as vertex decomposable complexes. They proved that the Stanley-Reisner ideal of every weakly vertex decomposable simplicial complex is Gorenstein linked to an ideal of indeterminates via a sequence of basic double links. In this paper, we explore basic double G-links between squarefree monomial ideals beyond the weakly vertex decomposable setting.

Our first contribution is a structural result about certain basic double links which involve an edge ideal. Specifically, suppose \(I(G)\) is the edge ideal of a graph G. When \(I(G)\) is a basic double link of a monomial ideal \(B\) on an arbitrary homogeneous ideal \(A\), we give a generating set for \(B\) in terms of \(G\) and show that this basic double link must be of degree \(1\). Our second focus is on examples from the literature of simplicial complexes known to be Cohen-Macaulay but not weakly vertex decomposable. We show that these examples are not basic double links of any other squarefree monomial ideals.
Differential operators, retracts, and toric face rings (with Christine Berkesch, C.-Y. Jean Chan, Laura Felicia Matusevich, Janet Page, and Janet Vassilev)

Algebra & Number Theory 17 (2023), no. 11, 1959–1984.
Journal version
arXiv:2112.00266

We give explicit descriptions of rings of differential operators of toric face rings in characteristic \(0\). For quotients of normal affine semigroup rings by radical monomial ideals, we also identify which of their differential operators are induced by differential operators on the ambient ring. Lastly, we provide a criterion for the Gorenstein property of a normal affine semigroup ring in terms of its differential operators.

Our main technique is to realize the \(k\)-algebras we study in terms of a suitable family of their algebra retracts in a way that is compatible with the characterization of differential operators. This strategy allows us to describe differential operators of any \(k\)-algebra realized by retracts in terms of the differential operators on these retracts, without restriction on char(\(k\)).
Bumpless pipe dreams encode Gröbner geometry of Schubert polynomials (with Anna Weigandt)

Submitted, 50 pages.
arXiv:2108.08370

In their study of infinite flag varieties, Lam, Lee, and Shimozono (2021) introduced bumpless pipe dreams in a new combinatorial formula for double Schubert polynomials. These polynomials are the \(T\times T\)-equivariant cohomology classes of matrix Schubert varieties and of their flat degenerations. We give diagonal term orders with respect to which bumpless pipe dreams index the irreducible components of diagonal Gröbner degenerations of matrix Schubert varieties, counted with scheme-theoretic multiplicity.

This indexing was conjectured by Hamaker, Pechenik, and Weigandt (2022). We also give a generalization to equidimensional unions of matrix Schubert varieties. This result establishes that bumpless pipe dreams are dual to and as geometrically natural as classical pipe dreams, for which an analogous anti-diagonal theory was developed by Knutson and Miller (2005).
An illustrated view of differential operators of a reduced quotient of an affine semigroup ring (with Christine Berkesch, C.-Y. Jean Chan, Laura Felicia Matusevich, Janet Page, and Janet Vassilev), 2021. In: Miller, C., Striuli, J., Witt, E.E. (eds) Women in Commutative Algebra. Association for Women in Mathematics Series, vol 29. Springer, Cham, 49–94.

Proceedings version
arXiv:2105.04074

Through examples, we illustrate how to compute differential operators on a quotient of an affine semigroup ring by a radical monomial ideal, when working over an algebraically closed field of characteristic \(0\).
Homological and combinatorial aspects of virtually Cohen–Macaulay sheaves (with Christine Berkesch, Michael C. Loper, and Jay Yang)

Transactions of the London Mathematical Society 8 (2021), no. 1, 413–434.
Journal version
arXiv:2012.14047

When studying a graded module \(M\) over the Cox ring of as mooth projective toric variety \(X\), there are two standard types of resolutions commonly used to glean information: free resolutions of M and vector bundle resolutions of its sheafification. Each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions can resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman, and Smith introduced virtual resolutions, which capture desirable geometric information and are also amenable to algebraic and combinatorial study. The theory of virtual resolutions includes a notion of a virtually Cohen–Macaulay property, though tools for assessing which modules are virtually Cohen–Macaulay have only recently started to be developed.

In this paper, we continue this research program in two related ways. The first is that, when \(X\) is a product of projective spaces, we produce a large new class of virtually Cohen–Macaulay Stanley–Reisner rings, which we show to be virtually Cohen–Macaulay via explicit constructions of appropriate virtual resolutions reflecting the underlying combinatorial structure. The second is that, for an arbitrary smooth projective toric variety \(X\), we develop homological tools for assessing the virtual Cohen–Macaulay property. Some of these tools give exclusionary criteria, and others are constructive methods for producing suitably short virtual resolutions. We also use these tools to establish relationships among the arithmetically, geometrically, and virtually Cohen–Macaulay properties.
Diagonal degenerations of matrix Schubert varieties

Algebraic Combinatorics 6 (2023), no. 4, 1073-1094.
Journal version
arXiv:2008.01717

Knutson and Miller (2005) established a connection between the anti-diagonal Gröbner degenerations of matrix Schubert varieties and the pre-existing combinatorics of pipe dreams. They used this correspondence to give a geometrically-natural explanation for the appearance of the combinatorially-defined Schubert polynomials as representatives of Schubert classes. Recently, Hamaker, Pechenik, and Weigandt (2022) proposed a similar connection between diagonal degenerations of matrix Schubert varieties and bumpless pipe dreams, newer combinatorial objects introduced by Lam, Lee, and Shimozono (2021). Hamaker, Pechenik, and Weigandt described new generating sets of the defining ideals of matrix Schubert varieties and conjectured a characterization of permutations for which these generating sets form diagonal Gröbner bases. They proved special cases of this conjecture and described diagonal degenerations of matrix Schubert varieties in terms of bumpless pipe dreams in these cases. The purpose of this paper is to prove the conjecture in full generality. The proof uses a connection between liaison and geometric vertex decomposition established in earlier work with Rajchgot (2021).
Geometric vertex decomposition and liaison (with Jenna Rajchgot)

Forum of Mathematics, Sigma 9 (2021), e70.
Journal version
arXiv:2005.14289

Geometric vertex decomposition and liaison are two frameworks that have been used to produce similar results about similar families of algebraic varieties. In this paper, we establish an explicit connection between these approaches. In particular, we show that each geometrically vertex decomposable ideal is linked by a sequence of elementary G-biliaisons of height 1 to an ideal of indeterminates and, conversely, that every G-biliaison of a certain type gives rise to a geometric vertex decomposition. As a consequence, we can immediately conclude that several well-known families of ideals are glicci, including Schubert determinantal ideals, defining ideals of varieties of complexes, and defining ideals of graded lower bound cluster algebras.
Toric double determinantal varieties (with Alexander Blose†, Owen McGrath†, and Jackson Morris†)

Communications in Algebra 49 (2021), no. 7, 3085–3093.
Journal version
arXiv:2006.04191

We examine Li's double determinantal varieties in the special case that they are toric. We recover from the general double determinantal varieties case, via a more elementary argument, that they are irreducible and show that toric double determinantal varieties are smooth. We use this framework to give a straighforward formula for their dimension. Finally, we use the smallest nontrivial toric double determinantal variety to provide some empirical evidence concerning an open problem in local algebra.
Gröbner bases and the Cohen–Macaulay property of Li's double determinantal varieties (with Nathan Fieldsteel)

Proceedings of the AMS, Series B, 7 (2020), 142–158.
Journal version
arXiv:1906.06817

We consider double determinantal varieties, a special case of Nakajima quiver varieties. Li conjectured that double determinantal varieties are normal, irreducible, Cohen–Macaulay varieties whose defining ideals have a Gröbner basis given by their natural generators. We use liaison theory to prove this conjecture in a manner that generalizes results for mixed ladder determinantal varieties. We also give a formula for the dimension of a double determinantal variety.
Characterizing finite length local cohomology in terms of bounds on Koszul cohomology

Journal of Algebra 543 (2020), 198–224.
Journal version
arXiv:1810.01359

Let \((R, m, \kappa)\) be a local ring. We give a characterization of \(R\)-modules \(M\) whose local cohomology is finite length up to some index in terms of asymptotic vanishing of Koszul cohomology on parameter ideals up to the same index. In particular, we show that a quasi-unmixed module \(M\) is asymptotically Cohen-Macaulay if and only if \(M\) is Cohen–Macaulay on the punctured spectrum if and only if \(\mbox{sup}\{\ell(H_i(f_1,...,f_d;M)) \mid \sqrt{f_1,\ldots,f_d} = m, i < d\} < \infty\) for \(d = \mbox{dim}(M) = \mbox{dim}(R)\).
Lech's inequality, the Stückrad–Vogel conjecture, and uniform behavior of Koszul homology (with Linquan Ma, Phạm Hùng Quý, Ilya Smirnov, and Yongwei Yao)

Advances in Mathematics 347 (2019), 442–472.
Journal version
arXiv:1808.01051

Let \((R,𝔪)\) be a Noetherian local ring, and let \(M\) be a finitely generated \(R\)-module of dimension \(d\). We prove that the set \(\{l(M/IM)e(I,M) \mid \sqrt{I}=𝔪\}\) is bounded below by \(1/d!e(\overline{R})\) where \(\overline{R} =R/Ann(M)\). Moreover, when \(\widehat{M}\) is equidimensional, this set is bounded above by a finite constant depending only on \(M\). The lower bound extends a classical inequality of Lech, and the upper bound answers a question of Stückrad–Vogel in the affirmative. As an application, we obtain results on uniform behavior of the lengths of Koszul homology modules.

Other writings:

The MatrixSchubert package for Macaulay2 (with Ayah Almousa, Sean Grate, Daoji Huang, Adam LaClair, Yuyuan Luo, and Joseph McDonough)

Journal of Software for Algebra and Geometry 15 (2025), 412–55.

This paper accompanies the MatrixSchubert package that we wrote for release in version 1.23 of Macaulay2, which is open source software used for computations in commutative algebra and algebraic geometry.
Bumpless pipe dreams encode Gröbner geometry of Schubert polynomials (with Anna Weigandt)

Séminaire Lotharingien de Combinatoire 86B (2022), Art. 84, 12pp.

FPSAC extended abstract (2022), accepted as talk.
Geometric vertex decomposition and liaison (with Jenna Rajchgot)

Séminaire Lotharingien de Combinatoire 85B (2021), Art. 13, 12pp.

FPSAC extended abstract (2021), accepted as poster.
Combinatorial aspects of virtually Cohen–Macaulay sheaves (with Christine Berkesch, Michael C. Loper, and Jay Yang)

Séminaire Lotharingien de Combinatoire 85B (2021), Art. 63, 13pp.

FPSAC extended abstract (2021), accepted as poster.
Relationships among Hilbert-Samuel multiplicities, Koszul cohomology, and local cohomology

Thesis, University of Michigan (2018).

(with Nate Kornell and Katherine Rawson)
Journal of Experimental Psychology: Learning, Memory, and Cognition, 41(1):283 (2015).

(with Nate Kornell and Veronica Rabelo)
Journal of Applied Research in Memory and Cognition, 1:259 (2015).