Finite $p$-groups of birational automorphisms and characterizations of rational varieties

TL;DR

This paper investigates the structure of finite p-subgroups within birational automorphism groups of rationally connected varieties, establishing conditions under which such varieties are rational based on the presence of large p-subgroups.

Contribution

It introduces a bound R(n) for p, linking the size of p-subgroups in birational automorphism groups to the rationality of varieties, leveraging boundedness of Fano varieties.

Findings

Existence of a constant R(n) for each dimension n

Rationality of varieties with large p-subgroups in automorphism groups

Applications to Jordan property in algebraic geometry

Abstract

We study finite $p$-subgroups of birational automorphism groups. By virtue of boundedness theorem of Fano varieties, we prove that there exists a constant $R(n)$ such that a rationally connected variety of dimension $n$ over an algebraically closed field is rational if its birational automorphism group contains a $p$-subgroups of maximal rank for $p > R(n)$. Some related applications on Jordan property are discussed.