On the connectedness of the automorphism group of an affine toric variety
Veronika Kikteva
TL;DR
The paper derives a combinatorial and divisor-class-theoretic criterion for the connectedness of the automorphism group of an affine toric variety, tying $\mathrm{Aut}(X)^0$ to the kernel of the action on $\mathrm{Cl}(X)$ and to the absence of nontrivial automorphisms of $\mathrm{Cl}(X)$ permuting invariant divisors. Using Cox rings and their $\mathrm{Cl}(X)$-grading, the authors show that for non-degenerate $X$ the component group $\mathrm{Aut}(X)/\mathrm{Aut}(X)^0$ is finite and described by $\mathrm{Aut}(\mathrm{Cl}(X)) \cap \Sigma_D$, with a universal bound of $r!$ where $r$ is the number of rays. The results yield a precise criterion for toric surfaces and a spectrum of examples, including cases with non-abelian component groups, illustrating both connected and non-connected automorphism groups. The work extends the understanding of automorphism groups from complete simplicial toric varieties to the affine setting and provides tools for explicit computation of the component structure via class groups and ray data.
Abstract
We obtain a criterion for the automorphism group of an affine toric variety to be connected in combinatorial terms and in terms of the divisor class group of the variety. The component group of the automorphism group of a non-degenerate affine toric variety is described. In particular, we show that the number of connected components of the automorphism group is finite.