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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.

##### MSC:
 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
##### Keywords:
hypergeometric system; semigroup ring; holomorphic D-module
Quasidegrees
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##### References:
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