Fano hypersurfaces with no finite order birational automorphisms

TL;DR

This paper investigates the properties of finite order birational automorphisms of hypersurfaces, proving that very general high-degree hypersurfaces lack such automorphisms using specialization techniques.

Contribution

It introduces a novel approach using specialization homomorphisms to analyze finite order birational automorphisms on hypersurfaces.

Findings

Very general hypersurfaces of degree d ≥ 5⌈(n+3)/6⌉ have no finite order birational automorphisms.

Finite order birational automorphisms cannot specialize to the identity in families over a DVR.

The method applies to automorphisms of order coprime to the residue characteristic.

Abstract

We use the specialization homomorphism for the birational automorphism group to study finite order birational automorphisms. For a family of varieties over a DVR, we prove that a birational automorphism of order coprime to the residue characteristic cannot specialize to the identity. As an application, we show that very general $n$-dimensional hypersurfaces of degree $d \geq 5 \lceil (n+3)/6 \rceil$ have no finite order birational automorphisms.