Automorphisms of Nonnormal Toric Varieties

This is a survey of recent results on multiple transitivity for automorphism groups of affine algebraic varieties. The property of infinite transitivity of the special automorphism group is treated, which is equivalent to the flexibility of the corresponding affine variety. These properties have important algebraic and geometric consequences. At the same time they are fulfilled for wide classes of varieties. Also, the situations are studied where infinite transitivity occurs for automorphism groups generated by finitely many one-parameter subgroups. In the appendices to the paper, the results on infinitely transitive actions in complex analysis and in combinatorial group theory are discussed.

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On the Automorphism Group of Non-Necessarily Normal Affine Toric Varieties

Díaz

Liendo

2023

International Mathematics Research Notices

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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 $X^{\prime}$ 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.

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Automorphisms of Nonnormal Toric Varieties

Cited by 3 publications

References 12 publications

Limit points and additive group actions

Limit points and additive group actions

Automorphisms of algebraic varieties and infinite transitivity

On the Automorphism Group of Non-Necessarily Normal Affine Toric Varieties

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