Borel subgroups of the plane Cremona group

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 auto-morphism group Aut(A(2)) of the affine plane. In this paper, we describe all Borel subgroups of the complex Cremona group Bir(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) admits Borel subgroups of any rank r is an element of {0, 1, 2} and that all Borel subgroups of rank r is an element of {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. 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 moduli space is of dimension 2g - 1.