Moduli Problems and Hilbert Schemes

Abstract

The starting point for the construction of moduli schemes or algebraic moduli spaces is A. Grothendieck’s theorem on the existence of Hilbert schemes, i.e. of schemes whose points classify closed subschemes of a projective space. Before recalling his results, let us make precise what we understand by a moduli functor, and let us recall D. Mumford’s definition of a coarse moduli scheme. We will state the results on the existence of moduli for different moduli problems of manifolds. As a very first step towards their proofs, we will discuss properties a reasonable moduli functor should have and we will apply them to show that the manifolds or schemes considered correspond to the points of a locally closed subscheme of a certain Hilbert scheme.