Weight polytopes and energy functionals of toric varieties

TL;DR

This paper establishes a geometric correspondence between the weight polytope of the Hurwitz form of a polarized smooth toric variety and the convex hull of characteristic vectors from regular triangulations, using K-energy slope formulas.

Contribution

It proves the equivalence of the weight polytope with a convex hull of characteristic vectors in the toric setting, connecting algebraic and combinatorial aspects.

Findings

Weight polytope matches convex hull of characteristic vectors.

Proof uses two slope formulas of K-energy in toric geometry.

Connects algebraic invariants with combinatorial triangulations.

Abstract

We prove that the weight polytope of the Hurwitz form of a polarized smooth toric variety coincides with the convex hull of the characteristic vectors introduced in the previous work of Ogusu and the author with respect to all regular triangulations of the momentum polytope. Our proof relies on the combination of the two slope formulas of K-energy in the toric setting.