On automorphism groups of smooth hypersurfaces

TL;DR

This paper characterizes smooth hypersurfaces in complex projective spaces with the largest possible automorphism groups, showing they are mostly Fermat hypersurfaces, and explicitly describes exceptions.

Contribution

It establishes a classification of hypersurfaces with maximal automorphism groups and explicitly details the exceptions and their automorphism groups.

Findings

Maximal automorphism groups correspond to Fermat hypersurfaces.

Explicit equations and automorphism groups are provided for exceptions.

Most smooth hypersurfaces with large automorphism groups are Fermat hypersurfaces.

Abstract

We show that smooth hypersurfaces in complex projective spaces with automorphism groups of maximum size are isomorphic to Fermat hypersurfaces, with a few exceptions. For the exceptions, we give explicitly the defining equations and automorphism groups.