Birational maps of Severi-Brauer surfaces, with applications to Cremona groups of higher rank

Abstract: Cremona groups are groups of birational transformations of a projective space. Their structure depends on the dimension and the field. In this talk, however, we will first focus on birational transformations of (non-trivial) Severi-Brauer surfaces, that is, surfaces that become isomorphic to the projective plane over the algebraic closure of K. Such surfaces do not contain any K-rational point. We will prove that if such a surface contains a point of degree 6, then its group of birational transformations is not generated by elements of finite order as it admits a surjective group homomorphism to the integers. As an application, we use this result to study Mori fiber spaces over the field of complex numbers, for which the generic fiber is a non-trivial Severi-Brauer surface. We prove that any group of cardinality at most the one of the complex numbers is a quotient of the Cremona group of rank 4 (and higher). This is joint work with Jérémy Blanc and Egor Yasinsky.

algebraic geometrycombinatorics

Audience: researchers in the topic

Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html