Homogeneous toric varieties

TL;DR

This paper characterizes toric varieties with transitive semisimple group actions, showing they are related to products of affine and projective spaces, using Cox realization and G-module analysis.

Contribution

It provides a classification of toric varieties admitting transitive semisimple group actions, linking them to specific product structures.

Findings

Toric varieties with transitive G-actions are between products of punctured affine spaces and projective spaces.

The Cox realization and G-module structure are key tools in the classification.

The paper offers a structural description of such toric varieties.

Abstract

A description of transitive actions of a semisimple algebraic group G on toric varieties is obtained. Every toric variety admitting such an action lies between a product of punctured affine spaces and a product of projective spaces. The result is based on the Cox realization of a toric variety as a quotient space of an open subset of a vector space V by a quasitorus action and on investigation of the G-module structure of V.