An approach of the Minimal Model Program for horospherical varieties via moment polytopes

TL;DR

This paper extends the Minimal Model Program to horospherical varieties by utilizing moment polytopes, generalizing known results from toric and spherical varieties to this broader class.

Contribution

It introduces a polytope-based approach to the MMP for horospherical varieties, completing and generalizing previous results from toric and spherical cases.

Findings

Generalizes MMP results from toric to horospherical varieties

Provides a polytope framework for MMP in horospherical varieties

Completes the MMP analysis for horospherical varieties

Abstract

We describe the Minimal Model Program in the family of $\mathbb{Q}$-Gorenstein projective horospherical varieties, by studying a family of polytopes defined from the moment polytope of a Cartier divisor of the variety we begin with. In particular, we generalize the results on MMP in toric varieties due to M. Reid, and we complete the results on MMP in spherical varieties due to M. Brion in the case of horospherical varieties.