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 kerℤ(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 Tor1 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≥ 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 q2, so this dimension is unbounded at fixed lattice rank.
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BibTeX
@misc{nakano2026obstruction,
title = {{The obstruction to intrinsic perturbation for $A$-hypergeometric series}},
author = {Ryunosuke Nakano},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.18006},
eprint = {2608.18006},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.18006},
}
RIS
TY - UNPB
TI - The obstruction to intrinsic perturbation for A-hypergeometric series
AU - Ryunosuke Nakano
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2608.18006
UR - https://arxiv.org/abs/2608.18006
AB - We show that, at a fake exponent of an A-hypergeometric system and for an ordered negative support family, intrinsic perturbation within ker_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 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≥ 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.
ER -