On the GKZ discriminant locus

TL;DR

The paper resolves the GKZ discriminant locus question by proving that the locus $V(A)$ defined as the union of face discriminants coincides with the zero set of the principal $A$-determinant $E_A$; moreover, $V(A)$ is already closed, so $V(A)=\overline{V(A)}=V(E_A)$. The approach blends toric geometry with a finiteness criterion: translating discriminant membership into finiteness of a graded module via a map $\gamma: \mathbb{C}[y_1,...,y_k]\to R$ and exploiting the toric stratification given by faces $F$ of $P$. The method relies on the toric variety $X_A$, its face-based orbits, and a base-change argument to extend local finiteness to global finiteness, using a Grothendieck-Serre-type argument. The results provide a clean, closure-free description of the GKZ discriminant, answer a question of Kite and Segal, and clarify how discriminants on faces assemble into the principal determinant.

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.

Theorems & Definitions (23)

• Proposition 1.1
• Remark 1.2
• Remark 1.3
• Proposition 2.1
• proof
• Proposition 2.2
• Lemma 3.1
• proof
• Corollary 3.2
• proof