Automorphisms of prime order of smooth cubic n-folds

TL;DR

This paper develops a criterion and computational method to classify smooth cubic hypersurfaces with prime order automorphisms, revealing bounds on the prime order and characterizing special cases like Klein n-folds.

Contribution

It introduces an effective criterion and computational approach for classifying automorphisms of prime order on smooth cubic hypersurfaces, including bounds and special cases.

Findings

Prime p is the order of an automorphism only if p<2^{n+1}.

Hypersurfaces with automorphism order p>2^n are isomorphic to Klein n-fold.

Complete classifications for automorphisms of prime order on cubic threefolds and fourfolds.

Abstract

In this paper we give an effective criterion as to when a prime number p is the order of an automorphism of a smooth cubic hypersurface of P^{n+1}, for a fixed n > 1. We also provide a computational method to classify all such hypersurfaces that admit an automorphism of prime order p. In particular, we show that p<2^{n+1} and that any such hypersurface admitting an automorphism of order p>2^n is isomorphic to the Klein n-fold. We apply our method to compute exhaustive lists of automorphism of prime order of smooth cubic threefolds and fourfolds. Finally, we provide an application to the moduli space of principally polarized abelian varieties.