Complex geometry seminar

The Cremona group of rank N over the field K is the group of birational transformations of the projective N-space that are defined over K. Cremona groups have been studied for a long time but especially the ones of rank 3 and higher remain mysterious, even over the complex numbers. Since 2019, we know that these have non-trivial normal subgroups, due to the construction of quotients by Blanc, Lamy and Zimmermann. In this talk, I will present the following result, obtained in joint work with Blanc and Yasinsky: “Let N be at least 4. Then any group (of cardinality at most the cardinality of the complex numbers) is a quotient of the complex Cremona group of rank N.” The proof uses the Sarkisov program from birational geometry, and Severi-Brauer surfaces from arithmetic geometry.