A geometric degree formula for $A$-discriminants and Euler obstructions of toric varieties

TL;DR

This paper provides explicit geometric formulas for the dimensions, degrees, and Euler obstructions of $A$-discriminant varieties and toric varieties, enhancing understanding of their algebraic and combinatorial properties.

Contribution

It introduces new explicit formulas for the degrees and dimensions of $A$-discriminants and Euler obstructions applicable to a broad class of toric varieties.

Findings

Formulas for dimensions and degrees of $A$-discriminant varieties.

Explicit combinatorial formulas for Euler obstructions of toric varieties.

Application of formulas to higher-codimensional cases.

Abstract

We give explicit formulas for the dimensions and the degrees of $A$-discriminant varieties introduced by Gelfand-Kapranov-Zelevinsky. Our formulas can be applied also to the case where the $A$-discriminant varieties are higher-codimensional and their degrees are described by the geometry of the configurations $A$. Moreover combinatorial formulas for the Euler obstructions of general (not necessarily normal) toric varieties will be also given.