Abelian Automorphism Groups of Quartic Surfaces and Cubic Fourfolds

TL;DR

This paper introduces a new classification method for abelian automorphism groups of hypersurfaces, successfully identifying such groups for smooth cubic fourfolds and quartic surfaces.

Contribution

The paper develops a novel approach to classify abelian automorphism groups of hypersurfaces and applies it to cubic fourfolds and quartic surfaces.

Findings

Classified abelian automorphism groups for smooth cubic fourfolds.

Classified abelian automorphism groups for quartic surfaces.

Established a new method for automorphism group classification.

Abstract

In this paper, we develop a new method to classify abelian automorphism groups of hypersurfaces. We use this method to classify (Theorem 4.2) abelian groups that admit a liftable action on a smooth cubic fourfold. A parallel result (Theorem 5.1) is obtained for quartic surfaces.