ABSTRACT
This chapter deals with the finer aspects of the geometry of algebraic varieties. It describes the first properties of morphisms, considering in particular the concepts of open and closed immersion and of finite morphism. The chapter demonstrates Chevalley's theorem that guarantees that morphisms are open with respect to the topology defined by the constructible sets. It defines the concept of complete variety generalizing projective varieties and projective spaces. This kind of varieties should be viewed as analogous in our category to compact topological spaces. The chapter explains the concepts of singular point and normal variety. It presents the definition and basic properties of the Proj variety associated to an affine graded k-algebra. The chapter talks about deeper into the geometric properties of varieties and morphisms. It introduces some basic definitions and results of the theory of schemes in order to present the valuative criterion of properness.
