Euler Discriminant of Complements of Hyperplanes

TL;DR

This paper investigates the Euler discriminant of hyperplane complements, proving it forms a hypersurface and providing explicit equations in generic and constrained cases, with combinatorial insights into component multiplicities.

Contribution

It establishes the hypersurface nature of the Euler discriminant for hyperplane complements and derives explicit defining equations for generic and special coefficient cases.

Findings

Euler discriminant is a hypersurface in coefficient space.

Explicit equations for the discriminant in generic and constrained cases.

Component multiplicities can be recovered combinatorially.

Abstract

The Euler discriminant of a family of very affine varieties is defined as the locus where the Euler characteristic drops. In this work, we study the Euler discriminant of families of complements of hyperplanes. We prove that the Euler discriminant is a hypersurface in the space of coefficients, and provide its defining equation in two cases: (1) when the coefficients are generic, and (2) when they are constrained to a proper subspace. In the generic case, we show that the multiplicities of the components can be recovered combinatorially. This analysis also recovers the singularities of an Euler integral. In the appendix, we discuss a relation to cosmological correlators.