Homological methods for hypergeometric families

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 $d$.