arXiv subjects
Publications and source records attributed to Christine Berkesch.
We introduce a variant of the Cox ring using $\mathbb{Q}$-Cartier divisors and use this to remedy various deficiencies of nonsimplicial toric varieties. Our main applications are: Cox's ideal-variety correspondence, an explicit classification of subschemes and sheaves in terms of multigraded modules, an associated toric Deligne-Mumford stack, and a stacky extension of the GKZ/Mori theory of toric varieties.
For any toric ideal $I$ in a polynomial ring $S$, we provide a combinatorial description of a free resolution of the integral closure of the $S$-module $S/I$. These new complexes arise from an extension of Bayer--Sturmfels' theory of cellular free resolutions. As applications, we unify several constructions for a resolution of the diagonal embedding of a toric variety, and compare the locally free resolutions for toric subvarieties introduced by Hanlon--Hicks--Lazarev and Brown--Erman.
We state and prove a realization of King's Conjecture for a category glued from the derived categories of all of the toric varieties arising from a given Cox ring. Our perspective extends ideas of Beilinson and Bondal to all semiprojective toric varieties.
For finite sets of points in $\mathbb{P}^n \times \mathbb{P}^m$, we produce short virtual resolutions, as introduced by Berkesch--Erman--Smith. We first intersect with a sufficiently high power of one set of variables for points in $\mathbb{P}^n \times \mathbb{P}^m$ to produce a virtual resolution of length $n+m$. Then, we describe an explicit virtual resolution of length 3 for a set of points in sufficiently general position in $\mathbb{P}^1 \times \mathbb{P}^2$, via a subcomplex of a free resolution. This first result generalizes to $\mathbb{P}^n \times \mathbb{P}^m$ work of Harada--Nowroozi--Van Tuyl, and the second partially generalizes work of Harada--Nowroozi--Van Tuyl and Booms-Peot, which were both for $\mathbb{P}^1 \times \mathbb{P}^1$. Along the way, we also note an explicit relationship between Betti numbers and higher difference matrices of bigraded Hilbert functions for $\mathbb{P}^n \times \mathbb{P}^m$.
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).
We provide a sharp upper bound on the quotient of the rank of an A-hypergeometric system with a three-dimensional torus action by the normalized volume of A; in this case, the upper bound is two.
The rank of an $A$-hypergeometric $D$-module $M_A(β)$, associated with a full rank $(d\times n)$-matrix $A$ and a vector of parameters $β\in \mathbb{C}^d$, is known to be the normalized volume of $A$, denoted $\mathrm{vol}(A)$, when $β$ lies outside the exceptional arrangement $\mathcal{E}(A)$, an affine subspace arrangement of codimension at least two. If $β\in \mathcal{E}(A)$ is simple, we prove that $d-1$ is a tight upper bound for the ratio $\mathrm{rank}(M_A(β))/\mathrm{vol}(A)$ for any $d\geq 3$. We also prove that the set of parameters $β$ such that this ratio is at least $2$ is an affine subspace arrangement of codimension at least $3$.
When studying a graded module $M$ over the Cox ring of a smooth 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.
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.
Syzygies capture intricate geometric properties of a subvariety in projective space. However, when the ambient space is a product of projective spaces or a more general smooth projective toric variety, minimal free resolutions over the Cox ring are too long and contain many geometrically superfluous summands. In this paper, we construct some much shorter free complexes that better encode the geometry.
We compute the $L$-characteristic cycle of an $A$-hypergeometric system and higher Euler-Koszul homology modules of the toric ring. We also prove upper semicontinuity results about the multiplicities in these cycles and apply our results to analyze the behavior of Gevrey solution spaces of the system.
We formalize, at the level of D-modules, the notion that A-hypergeometric systems are equivariant versions of the classical hypergeometric equations. For this purpose, we construct a functor on a suitable category of torus equivariant D-modules and show that it preserves key properties, such as holonomicity, regularity, and reducibility of monodromy representation. We also examine its effect on solutions, characteristic varieties, and singular loci. When applied to certain binomial D-modules, our functor produces saturations of the classical hypergeometric differential equations, a fact that sheds new light on the D-module theoretic properties of these classical systems.
We characterize the (regular) holonomicity of Horn systems of differential equations under a hypothesis that captures the most widely studied classical hypergeometric systems.
We apply the theory of Groebner bases to the computation of free resolutions over a polynomial ring, the defining equations of a canonically embedded curve, and the unirationality of the moduli space of curves of a fixed genus.
We consider integrals that generalize both the Mellin transforms of rational functions of the form 1/f and the classical Euler integrals. The domains of integration of our so-called Euler--Mellin integrals are naturally related to the coamoeba of f, and the components of the complement of the closure of the coamoeba give rise to a family of these integrals. After performing an explicit meromorphic continuation of Euler--Mellin integrals, we interpret them as A-hypergeometric functions and discuss their linear independence and relation to Mellin--Barnes integrals.
The main result is an elementary proof of holonomicity for A-hypergeometric systems, with no requirements on the behavior of their singularities, originally due to Adolphson [Ado94] after the regular singular case by Gelfand and Gelfand [GG86]. Our method yields a direct de novo proof that A-hypergeometric systems form holonomic families over their parameter spaces, as shown by Matusevich, Miller, and Walther [MMW05].
We discuss extremal Betti tables of resolutions in three different contexts. We begin over the graded polynomial ring, where extremal Betti tables correspond to pure resolutions. We then contrast this behavior with that of extremal Betti tables over regular local rings and over a bigraded ring.
We give a complete description of the cone of Betti diagrams over a standard graded hypersurface ring of the form k[x,y]/ , where q is a homogeneous quadric. We also provide a finite algorithm for decomposing Betti diagrams, including diagrams of infinite projective dimension, into pure diagrams. Boij--Soederberg theory completely describes the cone of Betti diagrams over a standard graded polynomial ring; our result provides the first example of another graded ring for which the cone of Betti diagrams is entirely understood.