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Homological methods for hypergeometric families. (English) Zbl 1095.13033
Let \(A\) be a \(d \times n\) integer matrix of rank \(d\) and \(\beta \in \mathbb C^d\) a complex vector. The \(A\)-hypergeometric system \(H_A(\beta)\) is a system of linear partial differential equations constructed from \(A\) and \(\beta\), which was introduced by I. M. Gel’fand, M. I. Graev and A. V. Zelevinskiĭ [Sov. Math., Dokl. 36, No.1, 5–10 (1987); translation from Dokl. Akad. Nauk SSSR 295, 14–19 (1987; Zbl 0661.22005)]. Assume that all columns of \(A\) lie in a single open halfspace. It is known that the rank of \(H_A(\beta)\), that is, the dimension of the space of holonomic solutions of \(H_A(\beta)\), is finite. Let \(\mathbb NA \subset \mathbb Z^d\) be the semigroup generated by the columns of \(A\). If the semigroup ring \(\mathbb C[\mathbb NA]\) is Cohen-Macaulay, the rank of \(H_A(\beta)\) is equal to \(\text{Vol}(A) := d! \times\) the volume of the convex hull of the columns of \(A\) and the origin \(O \in \mathbb R^d\). However, the rank of \(H_A(\beta)\) depends on the choice of \(\beta\) in general. We put \(\mathcal E_A = \{\beta \in \mathbb C^d \mid\) the rank of \(H_A(\beta) \neq \text{Vol}(A)\}\). In the present paper, the authors compute \(\mathcal E_A\) by using local cohomologies of the semigroup ring \(\mathbb C[\mathbb NA]\). They show that \(\mathcal E_A\) is equal to the closure of \(\{\alpha \in \mathbb Z^d \mid [\bigoplus_{p < d} H_{\mathfrak m}^p(\mathbb C[\mathbb NA])]_{-\alpha}\} \subset \mathbb C^d\) in the Zariski topology.

13N10 Commutative rings of differential operators and their modules
13D45 Local cohomology and commutative rings
33C70 Other hypergeometric functions and integrals in several variables
14M12 Determinantal varieties
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