The obstruction to intrinsic perturbation for $A$-hypergeometric series

Abstract

We show that, at a fake exponent of an $A$-hypergeometric system and for an ordered negative support family, intrinsic perturbation within $\ker_{\mathbb{Z}}(A)$ can produce a strictly smaller coefficient space than ambient perturbation, which answers a question of Okuyama--Saito. We measure the failure by an intrinsic-perturbation obstruction module, a quotient of two colon ideals whose graded dual is the ambient coefficient space modulo the intrinsic one, and we realize this module by two right-exact sequences involving $\operatorname{Tor}_1$ and present it finitely by two antichains. We give configurations of lattice rank 1 and 2 with nonzero obstruction, including one with an ordered distinguished collection consisting of all negative supports attained only by lattice shifts of nonnegative weight, and thereby settle the case left open by Okuyama--Saito. We give, for every integer $q\geq 1$, a configuration of lattice rank 2 with a connected column matroid and with a normal affine semigroup generated by the reduced Gröbner basis vectors, whose obstruction module has dimension $q^2$, so this dimension is unbounded at fixed lattice rank.