On the characterization of affine toric varieties by their automorphism group

TL;DR

This paper characterizes when an affine toric variety is uniquely identified by its automorphism group, revealing that only certain product structures allow such uniqueness, with optimal conditions established.

Contribution

It proves that affine toric varieties are uniquely determined by their automorphism groups under specific conditions, extending understanding of automorphism group characterizations.

Findings

Normal affine toric varieties are uniquely determined by their automorphism groups if they are not the algebraic torus.

The algebraic torus is characterized by automorphism groups within certain dimensional restrictions.

Examples show that the characterization does not extend beyond a certain dimension, establishing the result's optimality.

Abstract

In this paper we show that a normal affine toric variety X different from the algebraic torus is uniquely determined by its automorphism group in the category of affine irreducible, not necessarily normal, algebraic varieties if and only if X is isomorphic to the product of the affine line and another affine toric variety. In the case where X is the algebraic torus T, we reach the same conclusion if we restrict the category to only include irreducible varieties of dimension at most the dimension of T. There are examples of varieties of dimension one higher than T having the same automorphism group of T. Hence, this last result is optimal.