The Hilbert Scheme

Abstract

This chapter is a self-contained introduction to the Hilbert scheme. After a brief technical section on flatness, we construct the basic Hilbert scheme parameterizing subschemes of a fixed projective space with a prescribed Hilbert polynomial. We also introduce and discuss several variants of the basic construction, such as relative Hilbert schemes, flag Hilbert schemes, Hilbert schemes of morphisms, and so on. A sizable part of the chapter is devoted to the infinitesimal study of the Hilbert scheme and its variants, beginning with the classical theory of the characteristic system. We discuss and illustrate several pathological behaviors of Hilbert schemes, focussing especially on the case of Hilbert schemes of curves. In particular, we present Mumford’s example of a Hilbert scheme of space curves which is everywhere non-reduced along one of its components.