Combinatorics of rank jumps in simplicial hypergeometric systems
Abstract
Let A be an integer (d x n) matrix, and assume that the convex hull conv(A) of its columns is a simplex of dimension d-1. Write \NA for the semigroup generated by the columns of A. It was proved by M. Saito [math.AG/0012257] that the semigroup ring \CC[\NA] over the complex numbers \CC is Cohen-Macaulay if and only if the rank of the GKZ hypergeometric system H_A(beta) equals the normalized volume of conv(A) for all complex parameters beta in \CC^d. Our refinement here shows, in this simplicial case, that H_A(beta) has rank strictly larger than the volume of conv(A) if and only if beta lies in the Zariski closure in \CC^d of all \ZZ^d-graded degrees where the local cohomology H^i_m(\CC[\NA]) at the maximal ideal m is nonzero for some i < d.
Cite
@article{arxiv.math/0402071,
title = {Combinatorics of rank jumps in simplicial hypergeometric systems},
author = {Laura Felicia Matusevich and Ezra Miller},
journal= {arXiv preprint arXiv:math/0402071},
year = {2007}
}
Comments
6 pages, LaTeX