Borel subgroups of the plane Cremona group

TL;DR

This paper classifies Borel subgroups of the complex plane Cremona group, revealing their conjugacy properties and linking rank 0 cases to hyperelliptic curves with associated invariants.

Contribution

It provides a complete description of Borel subgroups in the Cremona group, showing their existence, conjugacy relations, and connection to hyperelliptic curves.

Findings

Borel subgroups exist for ranks 0, 1, 2

All rank 1 and 2 Borel subgroups are conjugate

Rank 0 Borel subgroups correspond to hyperelliptic curves with genus and moduli space invariants

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 ${{\rm Bir}({\mathbb P}^2)}$ up to conjugation, proving in particular that they are not necessarily conjugate. More precisely, we prove that ${{\rm Bir}({\mathbb P}^2)}$ 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 $g \geq 1$. 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 $2g-1$.