Automorphism Groups Of Smooth Cubic Threefolds

In this article,we classify groups which faithfully act on smooth cubic threefolds.It turns out that there are exactly 6 maximal ones(in particular,the automorphism group of any smooth cubic threefold must be isomorphic to a subgroup of one of these 6groups).We describe these 6 groups with explicit examples of target cubic threefolds.This article is divided into four chapters in total,and the central content is chapter three and chapter four.The first chapter is an introduction,which mainly introduces the background of the subject,the current status of the research,and some commonly used research methods in the research process.The second chapter is the basic knowledge of mathematics.It briefly introduces some basic theories involved in the research process of this article,mainly introduces the basic knowledge of group theory and the terms in the context of algebraic geometry and the definition of hypersurfaces on manifolds.The smoothness criterion,which plays an important role in our research,is introduced.The third chapter mainly introduces the concepts of smoothness and F-liftability,and explains how to use these two properties to carry out our research.In addition,we directly give 6 groups and the specific examples of the smooth cubic threefolds they act on.It turns out that these 6 groups are the 6 maximal groups mentioned above,and we will put the specific classification method in Chapter 4.The fourth chapter mainly studies the classification of the automorphism groups of smooth cubic threefolds.We first study all cyclic subgroups,secondly study all commutative subgroups.Then based on classification of finite groups of small order and the linear representation theory of finite groups,we study all Sylow p-subgroups with the help of computer program GAP.Finally we find all solvable subgroups and unsolvable subgroups of order 2~a3~b5~c.This completes the classification we want.