On the GKZ discriminant locus

TL;DR

This paper clarifies the structure of the GKZ discriminant locus, showing it is the union of all discriminant loci of faces of the convex hull, including higher codimension cases, without needing closures.

Contribution

It proves that the GKZ discriminant locus is exactly the union of all face discriminant loci, simplifying previous understanding and answering a question by Kite and Segal.

Findings

The GKZ discriminant locus equals the union of all face discriminant loci.

No closure operation is needed to describe the GKZ locus.

Includes discriminant loci of higher codimension faces.

Abstract

Let $A$ be an integral matrix and let $P$ be the convex hull of its columns. By a result of Gelfand, Kapranov and Zelevinski, the so-called principal $A$-determinant locus is equal to the union of the closures of the discriminant loci of the Laurent polynomials associated to the faces of $P$ that are hypersurfaces. In this short note we show that it is also the straightforward union of all the discriminant loci, i.e. we may include those of higher codimension, and there is no need to take closures. This answers a question by Kite and Segal.