On torus actions of higher complexity

TL;DR

This paper develops a systematic method to construct algebraic varieties with torus actions of higher complexity, extending previous work and classifying certain cases, using toric geometry and graded rings.

Contribution

It introduces a new construction approach for varieties with torus actions of higher complexity, including all Mori dream spaces with such actions, and classifies complexity two cases with low Picard number.

Findings

Constructed algebraic varieties with higher complexity torus actions.

Extended classification results to complexity two with Picard number ≤ 2.

Provided explicit descriptions using toric geometry and graded rings.

Abstract

We systematically produce algebraic varieties with torus action by constructing them as suitably embedded subvarieties of toric varieties. The resulting varieties admit an explicit treatment in terms of toric geometry and graded ring theory. Our approach extends existing constructions of rational varieties with torus action of complexity one and delivers all Mori dream spaces with torus action. We exhibit the example class of general arrangement varieties and obtain classification results in the case of complexity two and Picard number at most two, extending former work in complexity one.