A codimension formula for intrinsic perturbation of $A$-hypergeometric series

Abstract

We study the canonical formal solution space of a homogeneous $A$-hypergeometric system in a generic direction. We present as a finite-dimensional cokernel the quotient of that space by the span of the 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. We show that the presentation and the codimension transfer to the holomorphic solutions on a common nonsingular domain. We apply these results, for every integer $q\ge 1$, to a homogeneous $A$-hypergeometric system of lattice rank 2 with $5q$ columns. We show that the canonical formal solution space of this system has dimension $12q^2$ and that the span of the series obtained by intrinsic perturbation has codimension exactly $\lceil 3q^2/4\rceil$ in that space.

arXiv comment23 pages, 1 table; ancillary files: verification scripts for Sage and Macaulay2. Companion paper to arXiv:2608.18006, from which this paper was split; the obstruction module used here is constructed there. v2: the codimension for the family of lattice rank 2 is now computed exactly, and a comparison with arXiv:2609.02449 is added