arXiv subjects

Jenna Rajchgot

Publications and source records attributed to Jenna Rajchgot.

At least 19 recordsLinked to original sources

arXiv2025

Lifting Frobenius splittings through geometric vertex decomposition

Frobenius splitting, pioneered by Hochster and Roberts in the 1970s and Mehta and Ramanathan in the 1980s, is a technique in characteristic $p$ commutative algebra and algebraic geometry used to control singularities. In the aughts, Knutson showed that Frobenius splittings of a certain type descend through Gr\"obner degeneration of a certain type, called geometric vertex decomposition. In the present paper, we give a partial converse to Knutson's result. We show that a Frobenius splitting that compatibly splits both link and deletion of a geometric vertex decomposition can, under an additional hypothesis on the form of the splitting, be lifted to a splitting that compatibly splits the original ideal. We discuss an example showing that the additional hypothesis cannot be removed. Our argument uses the relationship between geometric vertex decomposition and Gorenstein liaison developed by Klein and Rajchgot. Additionally, we show that Li's double determinantal varieties defined by maximal minors are Frobenius split.

math.AC↗

arXiv2025

Multicomplex Configurations: a case study in Gorenstein Liaison

We introduce and investigate multicomplex configurations, a class of projective varieties constructed via specialization of the polarizations of Artinian monomial ideals. Building upon geometric polarization and geometric vertex decomposition, we establish conditions under which such configurations retain desirable algebraic properties. In particular, we show that, given suitable choices of linear forms for substitution, the resulting ideals admit Gr\"obner bases with prescribed initial ideals and are in the Gorenstein liaison class of a complete intersection.

math.AC↗

arXiv2025

Combinatorial proofs of the type A quiver component formulas

The K-theoretic quiver component formula expresses the K-polynomial of a type A quiver locus as an alternating sum of products of double Grothendieck polynomials. This formula was conjectured by A. Buch and R. Rim\'anyi and later proved by R. Kinser, A. Knutson, and the second author. We provide a new proof of this formula which replaces Gr\"obner degenerations by combinatorics. Along the way, we obtain a new proof of A. Buch and R. Rim\'anyi's cohomological quiver component formula. Again, our proof replaces geometric techniques by combinatorics.

math.CO↗

arXiv2025

A note on toric ideals of graphs and Knutson-Miller-Yong decompositions

We use a Gr\"obner basis technique first introduced by Knutson, Miller and Yong to study the interplay between properties of a graph $G$ and algebraic properties of the toric ideal that it defines. We first recover a well-known height formula for the toric ideal of a graph $I_G$ and demonstrate an algebraic property that can detect when a graph deletion is bipartite. We also bound the chromatic number $\chi(G)$ using information about an initial ideal of $I_G$.

math.AC↗

arXiv2024

Equivariant geometry of symmetric quiver orbit closures

We unify problems about the equivariant geometry of symmetric quiver representation varieties, in the finite type setting, with the corresponding problems for symmetric varieties $GL(n)/K$ where $K$ is an orthogonal or symplectic group. In particular, we translate results about singularities of orbit closures; combinatorics of orbit closure containment; and torus equivariant cohomology and K-theory between these classes of varieties. We obtain these results by constructing explicit embeddings with nice properties of homogeneous fiber bundles over type $A$ symmetric quiver representation varieties into symmetric varieties.

math.AG↗

arXiv2024

Polarization and Gorenstein liaison

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 $P(I)$, we give conditions that allow for the lifting of an associated basic double G-link of $P(I)$ 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 stable Cohen--Macaulay monomial ideals are vertex decomposable. We then introduce and study polarization of a Gr\"obner 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.

math.AC↗

arXiv2024

On basic double G-links of squarefree monomial ideals

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 G-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 G-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 G-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 G-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.

math.AC↗

arXiv2024

Gröbner geometry for regular nilpotent Hessenberg Schubert cells

A regular nilpotent Hessenberg Schubert cell is the intersection of a regular nilpotent Hessenberg variety with a Schubert cell. In this paper, we describe a set of minimal generators of the defining ideal of a regular nilpotent Hessenberg Schubert cell in the type $A$ setting. We show that these minimal generators are a Gröbner basis for an appropriate lexicographic monomial order. As a consequence, we obtain a new computational-algebraic proof, in type $A$, of Tymoczko's result that regular nilpotent Hessenberg varieties are paved by affine spaces. In addition, we prove that these defining ideals are complete intersections, are geometrically vertex decomposable, and compute their Hilbert series. We also produce a Frobenius splitting of each Schubert cell that compatibly splits all of the regular nilpotent Hessenberg Schubert cells contained in it. This work builds on, and extends, work of the second and third author on defining ideals of intersections of regular nilpotent Hessenberg varieties with the (open) Schubert cell associated to the Bruhat-longest permutation.

math.AG↗

arXiv2023

Three invariants of geometrically vertex decomposable ideals

We study three invariants of geometrically vertex decomposable ideals: the Castelnuovo-Mumford regularity, the multiplicity, and the $a$-invariant. We show that these invariants can be computed recursively using the ideals that appear in the geometric vertex decomposition process. As an application, we prove that the $a$-invariant of a geometrically vertex decomposable ideal is non-positive. We also recover some previously known results in the literature including a formula for the regularity of the Stanley--Reisner ideal of a pure vertex decomposable simplicial complex, and proofs that some well-known families of ideals are Hilbertian. Finally, we apply our recursions to the study of toric ideals of bipartite graphs. Included among our results on this topic is a new proof for a known bound on the $a$-invariant of a toric ideal of a bipartite graph.

math.AC↗

arXiv2022

Castelnuovo-Mumford regularity of ladder determinantal varieties and patches of Grassmannian Schubert varieties

We give degree formulas for Grothendieck polynomials indexed by vexillary permutations and $1432$-avoiding permutations via tableau combinatorics. These formulas generalize a formula for degrees of symmetric Grothendieck polynomials which appeared in previous joint work of the authors with Y. Ren and A. St. Dizier. We apply our formulas to compute Castelnuovo-Mumford regularity of classes of generalized determinantal ideals. In particular, we give combinatorial formulas for the regularities of all one-sided mixed ladder determinantal ideals. We also derive formulas for the regularities of certain Kazhdan-Lusztig ideals, including those coming from open patches of Schubert varieties in Grassmannians. This provides a correction to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri (2015).

math.CO↗

arXiv2022

Hadamard Products and Binomial Ideals

We study the Hadamard product of two varieties $V$ and $W$, with particular attention to the situation when one or both of $V$ and $W$ is a binomial variety. The main result of this paper shows that when $V$ and $W$ are both binomial varieties, and the binomials that define $V$ and $W$ have the same binomial exponents, then the defining equations of $V \star W$ can be computed explicitly and directly from the defining equations of $V$ and $W$. This result recovers known results about Hadamard products of binomial hypersurfaces and toric varieties. Moreover, as an application of our main result, we describe a relationship between the Hadamard product of the toric ideal $I_G$ of a graph $G$ and the toric ideal $I_H$ of a subgraph $H$ of $G$. We also derive results about algebraic invariants of Hadamard products: assuming $V$ and $W$ are binomial with the same exponents, we show that $\text{deg}(V\star W) = \text{deg}(V)=\text{deg}(W)$ and $\dim(V\star W) = \dim(V)=\dim(W)$. Finally, given any (not necessarily binomial) projective variety $V$ and a point $p \in \mathbb{P}^n \setminus \mathbb{V}(x_0x_1\cdots x_n)$, subject to some additional minor hypotheses, we find an explicit binomial variety that describes all the points $q$ that satisfy $p \star V = q\star V$.

math.AC↗

arXiv2022

Gröbner bases, symmetric matrices, and type C Kazhdan-Lusztig varieties

We study a class of combinatorially-defined polynomial ideals which are generated by minors of a generic symmetric matrix. Included within this class are the symmetric determinantal ideals, the symmetric ladder determinantal ideals, and the symmetric Schubert determinantal ideals of A. Fink, J. Rajchgot, and S. Sullivant. Each ideal in our class is a type C analog of a Kazhdan-Lusztig ideal of A. Woo and A. Yong; that is, it is the scheme-theoretic defining ideal of the intersection of a type C Schubert variety with a type C opposite Schubert cell, appropriately coordinatized. The Kazhdan-Lusztig ideals that arise are exactly those where the opposite cell is $123$-avoiding. Our main results include Gröbner bases for these ideals, prime decompositions of their initial ideals (which are Stanley-Reisner ideals of subword complexes) and combinatorial formulas for their multigraded Hilbert series in terms of pipe dreams.

math.AG↗

arXiv2022

Geometric vertex decomposition and liaison for toric ideals of graphs

The geometric vertex decomposability property for polynomial ideals is an ideal-theoretic generalization of the vertex decomposability property for simplicial complexes. Indeed, a homogeneous geometrically vertex decomposable ideal is radical and Cohen-Macaulay, and is in the Gorenstein liaison class of a complete intersection (glicci). In this paper, we initiate an investigation into when the toric ideal $I_G$ of a finite simple graph $G$ is geometrically vertex decomposable. We first show how geometric vertex decomposability behaves under tensor products, which allows us to restrict to connected graphs. We then describe a graph operation that preserves geometric vertex decomposability, thus allowing us to build many graphs whose corresponding toric ideals are geometrically vertex decomposable. Using work of Constantinescu and Gorla, we prove that toric ideals of bipartite graphs are geometrically vertex decomposable. We also propose a conjecture that all toric ideals of graphs with a square-free degeneration with respect to a lexicographic order are geometrically vertex decomposable. As evidence, we prove the conjecture in the case that the universal Gr\"obner basis of $I_G$ is a set of quadratic binomials. We also prove that some other families of graphs have the property that $I_G$ is glicci.

math.AC↗

arXiv2021

Geometric vertex decomposition and liaison

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.

math.AC↗

arXiv2021

Toric and tropical Bertini theorems in positive characteristic

We generalize the toric Bertini theorem of Fuchs, Mantova, and Zannier to positive characteristic. A key part of the proof is a new algebraically closed field containing the field \kk(t_1,\dots,t_d) of rational functions over an algebraically closed field \kk of prime characteristic. As a corollary, we extend the tropical Bertini theorem of Maclagan and Yu to arbitrary characteristic, which removes the characteristic dependence from the d-connectivity result for tropical varieties from that paper.

math.AG↗

arXiv2020

Degrees of symmetric Grothendieck polynomials and Castelnuovo-Mumford regularity

We give an explicit formula for the degree of the Grothendieck polynomial of a Grassmannian permutation and a closely related formula for the Castelnuovo-Mumford regularity of the Schubert determinantal ideal of a Grassmannian permutation. We then provide a counterexample to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri on a formula for regularities of standard open patches of particular Grassmannian Schubert varieties and show that our work gives rise to an alternate explicit formula in these cases. We end with a new conjecture on the regularities of standard open patches of arbitrary Grassmannian Schubert varieties.

math.CO↗

arXiv2020

Type D quiver representation varieties, double Grassmannians, and symmetric varieties

We unify aspects of the equivariant geometry of type $D$ quiver representation varieties, double Grassmannians, and symmetric varieties $GL(a+b)/GL(a)\times GL(b)$; in particular we translate results about singularities of orbit closures, combinatorics of orbit closure containment, and torus equivariant $K$-theory between these three families. These results are all obtained from our generalization of a construction of Zelevinsky for type $A$ quivers to the type $D$ setting. More precisely, we give explicit embeddings with nice properties of homogeneous fiber bundles over type $D$ quiver representation varieties into these symmetric varieties.

math.AG↗

arXiv2018

Three combinatorial formulas for type A quiver polynomials and K-polynomials

We provide combinatorial formulas for the multidegree and K-polynomial of an arbitrarily oriented type A quiver locus. These formulas are generalizations of three of Knutson-Miller-Shimozono's formulas from the equioriented setting; in particular, we prove the K-theoretic component formula conjectured by Buch and Rimányi.

math.AG↗