On lattice polarizable cubic fourfolds

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. Contents 1. Introduction 1 2. Notation 3 3. Partitionability and characteristic sets 4 4. Examples and reduction 10 5. Automorphism groups of cubic fivefolds 13 6. Automorphism groups of cubic fourfolds 19 References 25