Title: Finite order birational automorphisms of Fano hypersurfaces
Abstract: The birational automorphism group of an algebraic variety is an interesting birational invariant. For general type varieties this group is always finite, but for Fano varieties the situation is more complicated; for example, for P^n, the birational automorphism group is the mysterious Cremona group. In this talk, we study Fano hypersurfaces. We prove that there exist Fano hypersurfaces of arbitrarily high Fano index (in sufficiently high dimension) that admit no finite order birational automorphisms. A key input is the study of a specialization homomorphism for the birational automorphism group, which was defined by Matsusaka and Mumford. This work is joint with Nathan Chen and David Stapleton.