On the order of an automorphism of a smooth hypersurface

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)≠ (2,4), and (q,d)=(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=n+2(1-d) where n+2 is the (n+2)-th cyclotomic polynomial. Finally, we provide some applications to intermediate jacobians of Klein hypersurfaces.