Automorphisms of toric varieties and Gale duality

Ivan Arzhantsev; Kirill Shakhmatov

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arXiv · 2609.09051

Automorphisms of toric varieties and Gale duality

Ivan Arzhantsev·Kirill Shakhmatov

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Abstract

We classify complete toric threefolds $X$ such that the automorphism group $\text{Aut}(X)$ acts on $X$ with an open orbit whose complement does not contain a divisor. The latter condition means that for any ray $\rho$ of the fan $\Sigma_X$ there is a Demazure root of $\Sigma_X$ associated with $\rho$. We also find among these varieties those $X$ for which the group $\text{Aut}(X)$ is transitive on the smooth locus $X^{\text{reg}}$. The classifications are based on Gale-dual interpretations of these properties.

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Ivan Arzhantsev, Kirill Shakhmatov. 2026-09-08. Automorphisms of toric varieties and Gale duality. https://arxiv.org/abs/2609.09051

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