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Combinatorics of rank jumps in simplicial hypergeometric systems


Authors: Laura Felicia Matusevich and Ezra Miller
Journal: Proc. Amer. Math. Soc. 134 (2006), 1375-1381
MSC (2000): Primary 33C70; Secondary 14M25, 13N10, 13D45, 52B20, 13C14, 16S36, 20M25
DOI: https://doi.org/10.1090/S0002-9939-05-08245-6
Published electronically: November 17, 2005
MathSciNet review: 2199183
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Abstract: Let $ A$ be an integer $ d \times n$ matrix, and assume that the convex hull $ \operatorname{conv}(A)$ of its columns is a simplex of dimension $ d-1$ not containing the origin. It is known that the semigroup ring $ \mathbb{C}[\mathbb{N} A]$ is Cohen-Macaulay if and only if the rank of the GKZ hypergeometric system $ H_A(\beta)$ equals the normalized volume of $ \operatorname{conv}(A)$ for all complex parameters $ \beta \in \mathbb{C}^d$ (Saito, 2002). Our refinement here shows that $ H_A(\beta)$ has rank strictly larger than the volume of $ \operatorname{conv}(A)$ if and only if $ \beta$ lies in the Zariski closure (in  $ \mathbb{C}^d$) of all $ \mathbb{Z}^d$-graded degrees where the local cohomology $ \bigoplus_{i < d} H^i_\mathfrak{m}(\mathbb{C} [\mathbb{N}A])$ is nonzero. We conjecture that the same statement holds even when $ \operatorname{conv}(A)$ is not a simplex.


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Additional Information

Laura Felicia Matusevich
Affiliation: Mathematical Sciences Research Institute, Berkeley, California 94720
Address at time of publication: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: laura@math.tamu.edu

Ezra Miller
Affiliation: Mathematical Sciences Research Institute, Berkeley, California 94720
Address at time of publication: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 56267
Email: ezra@math.umn.edu

DOI: https://doi.org/10.1090/S0002-9939-05-08245-6
Received by editor(s): February 10, 2004
Received by editor(s) in revised form: December 3, 2004
Published electronically: November 17, 2005
Communicated by: Michael Stillman
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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