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 automorphism group of the affine plane. In this paper, we describe all Borel subgroups of the complex Cremona group 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 admits Borel subgroups of any rank and that all Borel subgroups of rank 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 . Hence, the conjugacy class of a rank 0 Borel subgroup admits two invariants: a discrete one, the genus , and a continuous one, corresponding to the coarse moduli space of hyperelliptic curves of genus . This moduli space is of dimension