Polyhedral Divisors and Algebraic Torus Actions

TL;DR

This paper introduces a comprehensive framework for describing normal affine varieties with algebraic torus actions using proper polyhedral divisors, extending classical cone constructions to a multigraded setting.

Contribution

It develops a unified theory that generalizes classical constructions of toric varieties to a broader class of algebraic varieties with torus actions.

Findings

Provides a complete description of such varieties using polyhedral divisors.

Extends classical cone constructions to the multigraded case.

Connects the theory of affine toric varieties with more general algebraic torus actions.

Abstract

We provide a complete description of normal affine varieties with effective algebraic torus action in terms of what we call proper polyhedral divisors on semiprojective varieties. Our theory extends classical cone constructions of Dolgachev, Demazure and Pinkham to the multigraded case, and it comprises the theory of affine toric varieties.