Automorphism groups of cubic fivefolds and fourfolds

TL;DR

This paper classifies the automorphism groups of smooth cubic fivefolds and fourfolds, introducing new concepts to understand their symmetries and identifying the maximal groups acting faithfully.

Contribution

It introduces notions of partitionability and characteristic sets, providing a complete classification of groups acting on cubic fivefolds and subgroups of automorphisms of fourfolds.

Findings

20 maximal groups acting on smooth cubic fivefolds identified

Complete classification of automorphism subgroups for smooth cubic fourfolds

New concepts of partitionability and characteristic sets introduced

Abstract

In this paper, we introduce notions of partitionability and characteristic sets of homogeneous polynomials and give a complete classification of groups faithfully acting on smooth cubic fivefolds. Specifically, we prove that there exist 20 maximal ones among all such groups. As an application, we classify all possible subgroups of the automorphism groups of smooth cubic fourfolds.