Hypergeometric Discriminants

TL;DR

This paper introduces a hypergeometric system linked to families of affine varieties, showing the Euler discriminant locus as its singular locus and providing a formula for hypergeometric discriminants in terms of likelihood equations.

Contribution

It establishes a new hypergeometric system framework for analyzing Euler discriminants in affine varieties and relates it to likelihood equations.

Findings

Euler discriminant locus is the singular locus of the hypergeometric system

The hypergeometric discriminant is characterized by a formula involving likelihood equations

The Euler discriminant locus is one-codimensional unless empty

Abstract

Given a family of varieties, the Euler discriminant locus distinguishes points where Euler characteristic differs from its generic value. We introduce a hypergeometric system associated with a flat family of very affine locally complete intersection varieties. It is proven that the Euler discriminant locus is its singular locus and is purely one-codimensional unless it is empty. Of particular interest is a family of very affine hypersurfaces. We coin the term hypergeometric discriminant for the characteristic cycle of the hypergeometric system and establish a formula in terms of likelihood equations.