Characteristic vectors for the Hurwitz polytopes of toric varieties

TL;DR

This paper introduces a new method using characteristic vectors related to regular triangulations to compute Hurwitz polytopes of smooth toric varieties, linking combinatorial and stability properties.

Contribution

It proposes a novel approach to compute Hurwitz polytopes via characteristic vectors and explores their relation to K-stability in toric geometry.

Findings

Convex hull of characteristic vectors is included in the Hurwitz polytope for smooth toric surfaces.

The introduced vectors relate to K-stability of pairs and toric K-stability.

The method provides a combinatorial tool for studying stability conditions in toric varieties.

Abstract

We introduce a characteristic vector with respect to a regular triangulation of the momentum polytope to compute the Hurwitz polytope of a given smooth toric variety. As a result, we prove that the convex hull of such vectors of all regular triangulations is included in the Hurwitz polytope of a smooth toric surface. In addition, we discuss the relations of such vectors to K-stability of pairs by Paul and toric K-stability by Donaldson.