Appears in collection : Summer School 2025 - Cremona Group

This is joint work with Andriy Regeta (University of Padova) and Christian Urech (ETH Z�rich). In the first part of the mini-course\, I will present structural results on algebraic families in the group Bir(X) of birational transformations of an irreducible variety X. In the second part\, I will discuss our recent progress in understanding Borel subgroups of Bir(X). We show that any Borel subgroup of Bir(X) has derived length at most twice the dimension of X\, with equality if and only if X is birationally equivalent to the projective space Pn and the Borel subgroup is conjugate to the standard Borel subgroup in the Cremona group Bir(Pn). In addition\, we construct examples of non-standard Borel subgroups of the Cremona group Bir(Pn) for n?2\, and of the affine Cremona group Aut(An) for n?3\, thereby settling conjectures of Popov and Furter-Poloni. I will also report on joint work with Christian Urech and Susanna Zimmermann (University of Basel) concerning the non-existence of Borel subgroups of derived length 2n-1 in Bir(Pn).