Spherical roots of spherical varieties

TL;DR

This paper generalizes Brion's theorem on the valuation cone of complex spherical varieties to fields with characteristic not equal to 2, extending the understanding of their geometric structure across different fields.

Contribution

It extends the fundamental theorem about valuation cones of spherical varieties to fields of arbitrary characteristic, including characteristic 2.

Findings

Valuation cone is a fundamental domain for a finite reflection group in characteristic not 2

A weaker version of the theorem holds in characteristic 2

Generalizes Akhiezer's classification of spherical rank-1 varieties

Abstract

Brion proved that the valuation cone of a complex spherical variety is a fundamental domain for a finite reflection group, called the little Weyl group. The principal goal of this paper is to generalize this fundamental theorem to fields of characteristic unequal to 2. We also prove a weaker version which holds in characteristic 2, as well. Our main tool is a generalization of Akhiezer's classification of spherical rank-1-varieties.