Combinatorics of rank jumps in simplicial hypergeometric systems
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- by Laura Felicia Matusevich and Ezra Miller PDF
- Proc. Amer. Math. Soc. 134 (2006), 1375-1381 Request permission
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.References
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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
- MR Author ID: 632562
- 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
- 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
- © Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2199183