Borel subgroups of the plane Cremona group

Abstract

It is well known that all Borel subgroups of a linear algebraic group are conjugate. This result also holds for the automorphism group $\mathrm{Aut}({\mathbb{A}}^2)$ of the affine plane \cite{BerestEshmatovEshmatov2016} (see also \cite{FurterPoloni2018}). In this paper, we describe all Borel subgroups of the complex Cremona group $\mathrm{Bir}({\mathbb{P}}^2)$ up to conjugation, proving in particular that they are not necessarily conjugate. More precisely, we prove that $\mathrm{Bir}({\mathbb{P}}^2)$ admits Borel subgroups of any rank $r\in \{0,1,2\}$ and that all Borel subgroups of rank $r\in \{1,2\}$ are conjugate. In rank $0$, there is a $1-1$ correspondence between conjugacy classes of Borel subgroups of rank $0$ and hyperelliptic curves of genus $g\ge 1$. Hence, the conjugacy class of a rank $0$ Borel subgroup admits two invariants: a discrete one, the genus $g$, and a continuous one, corresponding to the coarse moduli space of hyperelliptic curves of genus $g$. This latter space is of dimension $2g-1$.

arXiv comment43 pages, to appear in J. Reine Angew. Math

References in corpus (6)

• Topologies and structures of the Cremona groups
• On the inertia group of elliptic curves in the Cremona group of the plane
• On the geometry of the automorphism groups of affine varieties
• Elements and cyclic subgroups of finite order of the Cremona group
• Tori in the Cremona groups
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Cited by in corpus (1)

• Tits-type alternative for certain groups acting on algebraic surfaces