Introduction to the Theory of Moduli

1. Endomorphisms of vector spaces Throughout these notes, k is an algebraically closed field, varieties are reduced and irreducible k-schemes of finite type, and morphisms of varieties are k-morphisms. A moduli problem for a class of algebraic objects consists in two parts: finding the equivalence classes of the objects under a suitable equivalence relation (usually isomorphism), and parametrizing these classes by means of a scheme (or a geometric object of more general type). In this chapter we shall be interested in the moduli of endomorphisms of vector spaces. More precisely, let V be a vector space of dimension n over k, and let T be an endomorphism of V. The problem of classifying pairs (V, T) up to isomorphism is readily solved: there is a basis of V such that the matrix of T with respect to this basis is in the Jordan canonical form