On the order of an automorphism of a smooth hypersurface

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 ≥ 3, n ≥ 2, ( n, d ) ≠ (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 = Φ n +2 (1 − d ) where Φ n +2 is the ( n +2)-th cyclotomic polynomial. Finally, we provide some applications to intermediate jacobians of Klein hypersurfaces.