The automorphism group of a rigid affine variety

TL;DR

This paper investigates the automorphism groups of rigid affine varieties, proving the existence of a unique maximal torus and characterizing the automorphism group structure, especially for rigid trinomial affine hypersurfaces.

Contribution

It establishes the existence and uniqueness of a maximal torus in the automorphism group of rigid affine varieties and describes the automorphism groups of rigid trinomial hypersurfaces.

Findings

The automorphism group contains a unique maximal torus.

If the grading is pointed, the automorphism group is a finite extension of the torus.

Automorphisms of rigid trinomial hypersurfaces are fully described.

Abstract

An irreducible algebraic variety $X$ is rigid if it admits no nontrivial action of the additive group of the ground field. We prove that the automorphism group $\text{Aut}(X)$ of a rigid affine variety contains a unique maximal torus $\mathbb{T}$. If the grading on the algebra of regular functions $\mathbb{K}[X]$ defined by the action of $\mathbb{T}$ is pointed, the group $\text{Aut}(X)$ is a finite extension of $\mathbb{T}$. As an application, we describe the automorphism group of a rigid trinomial affine hypersurface and find all isomorphisms between such hypersurfaces.