On the order of an automorphism of a smooth hypersurface

TL;DR

This paper provides an effective criterion for determining when a positive integer q is the order of an automorphism of a smooth hypersurface, with specific results for prime orders and applications to Klein hypersurfaces.

Contribution

It introduces a complete criterion for automorphism orders of smooth hypersurfaces, especially for prime orders, and characterizes Klein hypersurfaces with automorphisms of large prime order.

Findings

Automorphism order q relates to divisibility conditions with d and d-1.

Prime order automorphisms p satisfy p<(d-1)^{n+1}.

Large prime automorphisms imply the hypersurface is Klein.

Abstract

In this paper we give an effective criterion as to when a positive integer q is the order of an automorphism of a smooth hypersurface of dimension n and degree d, for every d>2, n>1, (n,d)\neq (2,4), and \gcd(q,d)=\gcd(q,d-1)=1. This allows us to give a complete criterion in the case where q=p is a prime number. In particular, we show the following result: If X is a smooth hypersurface of dimension n and degree d admitting an automorphism of prime order p then p<(d-1)^{n+1}; and if p>(d-1)^n then X is isomorphic to the Klein hypersurface, n=2 or n+2 is prime, and p=\Phi_{n+2}(1-d) where \Phi_{n+2} is the (n+2)-th cyclotomic polynomial. Finally, we provide some applications to intermediate jacobians of Klein hypersurfaces.