About

Christine Berkesch works on homological questions about structure resulting from a group action. When a group acts on an algebraic variety, the combinatorics of the orbit structure and induced grading on the structure sheaf shed light on the geometry of the variety, by providing more tools for computing important algebro-geometric invariants. She earned her PhD in Mathematics from Purdue University in 2010 and her BA in Mathematics from Butler University in 2004. She is a professor at the School of Mathematics at the University of Minnesota Twin Cities, with research interests centered on the interplay between group actions, algebraic varieties, and their geometric and algebraic invariants.

Research topics

• Pure mathematics
• Mathematics
• Computer Science
• Artificial Intelligence
• Mathematical analysis
• Programming language
• Discrete mathematics
• Chemistry
• Physics

Selected publications

• On virtual resolutions of points in a product of two projective spaces

  Journal of Pure and Applied Algebra · 2025-05-16

  articleCorresponding
  PublisherDOI

• Cellular free resolutions for normalizations of toric ideals

  arXiv (Cornell University) · 2025-12-19

  preprintOpen access1st authorCorresponding

  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.

  PublisherDOI

• Cellular free resolutions for normalizations of toric ideals

  ArXiv.org · 2025-12-19

  articleOpen access1st authorCorresponding

  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.

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• King's Conjecture and the Cox category

  arXiv (Cornell University) · 2024-12-30

  preprintOpen access

  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.

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• On virtual resolutions of points in a product of projective spaces

  arXiv (Cornell University) · 2024-02-19

  preprintOpen access

  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 $\mathbb{P}^n \times \mathbb{P}^m$.

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• A sharp bound for hypergeometric rank in dimension three

  Selecta Mathematica · 2024-10-15

  article1st authorCorresponding
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• Differential operators, retracts, and toric face rings

  Algebra & Number Theory · 2023-10-03 · 1 citations

  articleOpen access1st authorCorresponding

  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).

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• A sharp bound for hypergeometric rank in dimension three

  arXiv (Cornell University) · 2023-01-11

  preprintOpen access1st authorCorresponding

  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.

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• Characteristic cycles and Gevrey series solutions of A -hypergeometric systems

  idUS (Universidad de Sevilla) · 2022-11-08 · 2 citations

  articleOpen access1st authorCorresponding

  We compute the 
\nL
\n-characteristic cycle of an 
\nA
\n-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.

  Publisher

• On the rank of an A$A$‐hypergeometric D$D$‐module versus the normalized volume of A$A$

  Bulletin of the London Mathematical Society · 2022-02-01 · 1 citations

  articleOpen access1st authorCorresponding

  The rank of an A $A$ -hypergeometric D $D$ -module M A ( β ) $M_A(\beta )$ , associated with a full-rank ( d × n ) $(d\times n)$ -matrix A $A$ and a vector of parameters β ∈ C d $\beta \in {\mathbb {C}}^d$ , is known to be the normalized volume of A $A$ , denoted vol ( A ) ${\operatorname{vol}}(A)$ , when β $\beta$ lies outside the exceptional arrangement E ( A ) ${\mathcal {E}}(A)$ , an affine subspace arrangement of codimension at least two. If β ∈ E ( A ) $\beta \in {\mathcal {E}}(A)$ is simple, we prove that d − 1 $d-1$ is a tight upper bound for the ratio rank ( M A ( β ) ) / vol ( A ) ${\operatorname{rank}}(M_A(\beta ))/{\operatorname{vol}}(A)$ for any d ⩾ 3 $d\geqslant 3$ . We also prove that the set of parameters β $\beta$ such that this ratio is at least two is an affine subspace arrangement of codimension at least three.

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Recent grants

• Multigraded Methods for Syzygies, Arrangements, and Differential Operators

  NSF · $274k · 2020–2024

• Homological Commutative Algebra and Group Actions in Geometry

  NSF · $162k · 2017–2020

• Homological commutative algebra, polyhedral structure, and algebraic geometry

  NSF · $104k · 2013–2014

• International Research Fellowship Program: The Solution Space of a Hypergeometric System

  NSF · $91k · 2010–2012

• Homological commutative algebra, polyhedral structure, and algebraic geometry

  NSF · $104k · 2013–2017

Frequent coauthors

• Laura Felicia Matusevich

  Texas A&M University

  24 shared

• Steven V Sam

  University of California, San Diego

  21 shared

• Daniel Erman

  University of Hawaiʻi at Mānoa

  16 shared

• Manoj Kummini

  15 shared

• Uli Walther

  11 shared

• Jens Forsgård

  9 shared

• Patricia Klein

  8 shared

• C-Y. Jean Chan

  6 shared