Gauss-Manin systems of families of Laurent polynomials and A-hypergeometric systems

TL;DR

This paper explores the structure of Gauss-Manin systems associated with Laurent polynomials, establishing a connection with A-hypergeometric systems through a four-term exact sequence and analyzing related extension groups.

Contribution

It introduces a natural four-term exact sequence linking Gauss-Manin systems and A-hypergeometric systems, providing new insights into their extension groups and non-splitting properties.

Findings

Established a four-term exact sequence for Gauss-Manin systems

Computed Ext and Tor groups for A-hypergeometric systems

Proved the sequence does not split

Abstract

In this note we study families of Gauss-Manin systems arising from Laurent polynomials with parametric coefficients under projection to the parameter space. For suitable matrices of exponent vectors, we exhibit a natural four-term exact sequence for which we then give an interpretation via generalized A-hypergeometric systems. We determine the extension groups from the parameter sheaf to the middle term of this sequence and show that the four-term sequence does not split. Auxiliary results include the computation of Ext and Tor groups of A-hypergeometric systems against the parameter sheaf.