Obstructions 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 gap 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. For a homogeneous system, we present as a finite-dimensional cokernel the quotient of the canonical formal solution space by the span of the canonical series obtained by intrinsic perturbation, taken over all exponents occurring in that space and all ordered negative support families, and we compute the codimension of that span; the presentation and the codimension transfer to the holomorphic solutions on a common nonsingular domain. We give configurations of lattice rank 1 and 2 with nonzero obstruction, one of which has an ordered distinguished collection and settles 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 Gr\"obner basis vectors, whose obstruction module has dimension $q^2$ and for which the sums of the canonical series obtained by intrinsic perturbation span a subspace of the holomorphic solution space of codimension at least $\lceil 3q^2/4\rceil$, so this codimension is unbounded at fixed lattice rank.