Characterization of affine surfaces with a torus action by their automorphism groups

TL;DR

This paper investigates how the automorphism groups of affine surfaces and varieties determine their structure, showing that for certain classes, the automorphism group uniquely characterizes the surface or variety.

Contribution

It proves that automorphism groups determine the structure of affine surfaces and toric varieties, extending known results to higher dimensions and non-toric cases.

Findings

Automorphism groups of affine surfaces determine their structure.

Complex affine toric surfaces are characterized by their automorphism groups.

Higher-dimensional affine toric varieties are uniquely identified by their automorphism groups.

Abstract

In this paper we prove that if two normal affine surfaces $S$ and $S'$ have isomorphic automorphism groups, then every connected algebraic group acting regularly and faithfully on $S$ acts also regularly and faithfully on $S'$. Moreover, if $S$ is non-toric, we show that the dynamical type of a 1-torus action is preserved in presence of an additive group action. We also show that complex affine toric surfaces are determined by the abstract group structure of their regular automorphism groups in the category of complex normal affine surfaces using properties of the Cremona group. As a generalization to arbitrary dimensions, we show that complex affine toric varieties, with the exception of the algebraic torus, are uniquely determined in the category of complex affine normal varieties by their automorphism groups seen as ind-groups.