Math @ Duke

Publications [#243927] of Ezra Miller
Papers Published
 with Matusevich, LF; Miller, E, Combinatorics of rank jumps in simplicial hypergeometric systems,
Proceedings of the American Mathematical Society, vol. 134 no. 5
(2006),
pp. 13751381, ISSN 00029939 [MR2006j:33016], [math.AC/0402071], [doi]
(last updated on 2018/03/18)
Abstract: Let A be an integer d × n matrix, and assume that the convex hull conv(A) of its columns is a simplex of dimension d  1 not containing the origin. It is known that the semigroup ring ℂ[Ndbl;A] is CohenMacaulay if and only if the rank of the GKZ hypergeometric system H A(β) equals the normalized volume of conv(A) for all complex parameters β ε ℂ d (Saito, 2002). Our refinement here shows that H A(β) has rank strictly larger than the volume of conv(A) if and only if β lies in the Zariski closure (in ℂ d) of all Zdbl; dgraded degrees where the local cohomology ⊕ i<d H mi(ℂ[ℕA]) is nonzero. We conjecture that the same statement holds even when conv(A) is not a simplex. © 2005 American Mathematical Society.


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