Borel subgroups of the plane Cremona group

Abstract It is well known that all Borel subgroups of a linear algebraic group are conjugate. Berest, Eshmatov, and Eshmatov have shown that this result also holds for the automorphism group Aut ⁢ ( 𝔸 2 ) {{\mathrm{Aut}}({\mathbb{A}}^{2})} of the affine plane. In this paper, we describe all Borel subgroups of the complex Cremona group Bir ⁢ ( ℙ 2 ) {{\rm Bir}({\mathbb{P}}^{2})} up to conjugation, proving in particular that they are not necessarily conjugate. In principle, this fact answers a question of Popov. More precisely, we prove that Bir ⁢ ( ℙ 2 ) {{\rm Bir}({\mathbb{P}}^{2})} admits Borel subgroups of any rank r ∈ { 0 , 1 , 2 } {r\in\{0,1,2\}} and that all Borel subgroups of rank r ∈ { 1 , 2 } {r\in\{1,2\}} are conjugate. In rank 0, there is a one-to-one correspondence between conjugacy classes of Borel subgroups of rank 0 and hyperelliptic curves of genus ℊ ≥ 1 {\mathcal{g}\geq 1} . Hence, the conjugacy class of a rank 0 Borel subgroup admits two invariants: a discrete one, the genus ℊ {\mathcal{g}} , and a continuous one, corresponding to the coarse moduli space of hyperelliptic curves of genus ℊ {\mathcal{g}} . This moduli space is of dimension 2 ⁢ ℊ - 1 {2\mathcal{g}-1} .