Abstract:
A degree $d \le n$ hypersurface in $P^n$ is a natural example of a higher-dimensional Fano variety. For a very general $X$ (with $n \ge 2, d \ge 3$) the group $\operatorname{Aut}(X)$ is trivial, but $\operatorname{Bir}(X)$ is a mysterious object. Using degenerations to finite characteristic fields Chen, Ji, and Stapleton recently proved that for $d \ge 5 * \textrm{ceil}(n/6 + 1/2)$ the group $\operatorname{Bir}(X)$ has no elements of finite order. The degeneration technique is a common method to understand the birational properties of hypersurfaces, but tracking the behavior of $\operatorname{Bir}(X)$ and its elements during the degeneration is often challenging. I will explain how this was done in that paper.
