Borel subgroups of the automorphism groups of affine toric surfaces

Ivan Arzhantsev; Mikhail Zaidenberg

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arXiv · 2507.09679

Borel subgroups of the automorphism groups of affine toric surfaces

Ivan Arzhantsev·Mikhail Zaidenberg

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Abstract

In [I. Arzhantsev and M. Zaidenberg, Acyclic curves and group actions on affine toric surfaces. Affine Algebraic Geometry, 1--41. World Scientific Publishing Co. 2013] we described the automorphism groups of the cyclic quotients of the affine plane. In this article, we study the Borel subgroups and, more generally, the maximal solvable subgroups of these ind-groups. We show that the cyclic quotients of the affine plane are divided into two species. In one of them, the Borel subgroups form a single conjugacy class, while in the other, there are two conjugacy classes of Borel subgroups. The proofs explore the Bass-Serre-Tits theory of groups acting on trees.

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Ivan Arzhantsev, Mikhail Zaidenberg. 2025-07-13. Borel subgroups of the automorphism groups of affine toric surfaces. https://arxiv.org/abs/2507.09679

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