Andriy Regeta - University of Jena
A characterization of rationality and Borel subgroups in the Cremona group

Abstract: In this talk I will present the following two results about the group of birational transformations (which we denote by Bir(X)) of an irreducible variety X:
The first one: If  P^n is a projective space and if Bir(X) is isomorphic to Bir(P^n), than X is rational.
Regarding the second one: it is well-known that Borel subgroups of an algebraic group (over algebraically closed fields) are conjugate. This is not the case for Bir(P^2). Nevertheless, all Borel subgroups of Bir(P^2) were classified  by J.-P. Furter and I. Heden. In the general case we show that a Borel subgroup of Bir(X) has derived length at most twice the dimension of X and in the case of equality X is rational, and the Borel subgroup is conjugate to the standard Borel subgroup in Bir(P^n). Moreover, we provide examples of Borel subgroups in Bir(P^n) of derived length strictly smaller than 2n for any n≥2.  This confirms the conjectures by  Popov and by Furter-Heden. We also show that the automorphism group Aut( A^n) of an affine space A^n contains non-conjugate Borel subgroups of n>2 and provide some structure results concerning the Borel subgroups of  Aut(A^n).
This talk is based on joint work with Christian Urech and Immanuel van Santen.