On the automorphism group of non-necessarily normal affine toric varieties

TL;DR

This paper characterizes when a non-normal affine toric surface shares its automorphism group with a normal one, revealing unique properties of the affine plane and classifying additive group actions on such varieties.

Contribution

It provides a classification of normalized additive group actions on non-normal affine toric varieties and characterizes automorphism groups of these surfaces.

Findings

Existence of a non-normal affine toric surface with the same automorphism group as a given surface, except for the affine plane.

Classification of normalized additive group actions via homogeneous locally nilpotent derivations.

General classification of homogeneous locally nilpotent derivations on semigroup algebras.

Abstract

Our main result is the following: let X be a normal affine toric surface without torus factor. Then there exists a non-normal affine toric surface X' with automorphism group isomorphic to the automorphism group of X if and only if X is different from the affine plane. As a tool, we first provide a classification of normalized additive group actions on a non-necessarily normal affine toric variety X of any dimension. Recall that normalized additive group actions on X are in correspondence with homogeneous locally nilpotent derivations on the algebra of regular functions of X. More generally, we provide a classification of homogeneous locally nilpotent derivations on the semigroup algebra of a commutative cancellative monoid.