A relation between number of integral points, volumes of faces and degree of the discriminant of smooth lattice polytopes

TL;DR

This paper derives a formula linking the degree of the discriminant of smooth toric varieties to the count of interior lattice points in dilated faces of the associated polytope, revealing geometric-combinatorial relationships.

Contribution

It introduces a novel formula connecting discriminant degree with lattice point counts in face dilations of smooth lattice polytopes.

Findings

Formula for discriminant degree in terms of interior lattice points

Connection between face volumes and discriminant properties

Enhanced understanding of lattice polytope geometry

Abstract

We present a formula for the degree of the discriminant of a smooth projective toric variety associated to a lattice polytope P, in terms of the number of integral points in the interior of dilates of faces of dimension greater or equal than $\lceil \frac {\dim P} 2 \rceil$.