A超几何级数的内在扰动障碍
The obstruction to intrinsic perturbation for $A$-hypergeometric series
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中文总结 AI 辅助
该研究针对A超几何系统的内在扰动问题，构造了内在扰动障碍模，计算了相关余维数，解决了奥山－斋藤留下的开放情形，且在固定格秩下余维数无界。
中文摘要 AI 辅助
奥山－斋藤（Okuyama--Saito）提出问题：对于A超几何系统的一个虚假指数处的有序负支撑族，关系格内部的内在扰动是否能产生比环境扰动严格更小的系数空间。答案是肯定的，我们通过内在扰动障碍模来衡量该差距，该模是两个冒号理想的商，其分次对偶是环境系数空间模以内在系数空间。我们通过涉及Tor₁的两个右正合序列实现该模，并通过两个反链有限表示。对于齐次系统，我们将典范形式解空间模以内在级数的张成表示为有限维余核，其中内在级数取自典范形式解空间中出现的所有虚假指数及所有有序负支撑族，并计算该张成的余维数；该表示和余维数可转移到公共非奇异域上的全纯解。具有非零障碍的格秩1和2的构型包括一个其 distinguished collection 有序的构型；该例子解决了奥山－斋藤留下的开放情形。对于每个整数q≥1，存在一个格秩2的构型，其具有连通列拟阵和由约化格罗比纳基向量生成的正规仿射半群，其障碍模的维数为q²，且其内在全纯解的余维数至少为⌈3q²/4⌉；在固定格秩下，该余维数是无界的。
英文摘要
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.
发表机构
- Graduate School of Science, Hokkaido University（北海道大学大学院理学研究院）
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