Hilbert scheme of d points on a smooth surface

Let S be a smooth algebraic surface. The Hilbert scheme of d points on S is the moduli space of length d subschemes of S.

Properties

 * Hilb(d)S is a smooth variety. (This is not true for Hilbert schemes of points on singular surfaces or higher-dimensional smooth varieties.)

Universal family
Since the Hilbert scheme is a fine moduli space for the moduli problem of families of length d subschemes of S, there is a universal family P (finite flat of degree d over Hilb(d)S), with a universal map P -> X that is an embedding on fibers of P -> Hilb(d)S.

Natural sheaves and bundles

 * There is a natural rank d vector bundle (name?) on Hilb(d)S whose fiber over a subscheme P is the vector space of regular functions on P. Precisely, this is the pushforward of OP to Hilb(d)S.

Natural maps to other objects
There is a natural map called the Hilbert-Chow morphism Hilb(d)S -> SymdS that records the reduced points of a length d subscheme, with multiplicity given by the length at each point.

Correspondences
There are no maps relating Hilb(d)S for various d (though there are rational maps Hilb(d)S x Sm -> Hilb(d+m)S). Instead, there are correspondences M_k \subseteq Hilb(d)S x Hilb(d+k)S for all k, where M_k consists of pairs (P_1,P_2) with P_1 a subscheme of P_2.

Natural subvarieties

 * For any point s\in S, there is a closed subvariety of Hilb(d)S consisting of subschemes supported at s. This is called the punctual Hilbert scheme Hilb(d)pun,2, and it is independent of both the point s and the ambient surface S, up to isomorphism.

Special cases

 * Hilbert scheme of d points on C2
 * Hilbert scheme of d points on P2
 * Hilbert scheme of d points on a K3 surface

Generalizations

 * Hilbert scheme of d points on a surface