Homological methods for hypergeometric families

Abstract:

We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rank-jumps in this general setting. Then we investigate rank-jump behavior for hypergeometric systems $H_A(\beta )$ arising from a $d \times n$ integer matrix $A$ and a parameter $\beta \in \mathbb {C}^d$. To do so we introduce an Euler–Koszul functor for hypergeometric families over $\mathbb {C}^d$, whose homology generalizes the notion of a hypergeometric system, and we prove a homology isomorphism with our general homological construction above. We show that a parameter $\beta \in \mathbb {C}^d$ is rank-jumping for $H_A(\beta )$ if and only if $\beta$ lies in the Zariski closure of the set of $\mathbb {C}^d$-graded degrees $\alpha$ where the local cohomology $\bigoplus _{i < d} H^i_\mathfrak m(\mathbb {C}[\mathbb {N} A])_\alpha$ of the semigroup ring $\mathbb {C}[\mathbb {N} A]$ supported at its maximal graded ideal $\mathfrak m$ is nonzero. Consequently, $H_A(\beta )$ has no rank-jumps over $\mathbb {C}^d$ if and only if $\mathbb {C}[\mathbb {N} A]$ is Cohen–Macaulay of dimension $d$.