of the Hilbert scheme is naturally isomorphic to the set of closed subschems of Pn×S that are flat over S. The closed subschems of Pn×S that are flat over S can informally be thought of as the families of subschemes of projective space parameterized by S. The Hilbert scheme HilbPn breaks up as a disjoint union pieces HilbPnP corresponding to the Hilbert polynomial of the subschemes of projective space with Hilbert polynomial P. Each of these pieces is projective over Spec(Z).
If X is a subscheme of n-dimensional projective space, then X corresponds to a graded ideal IX of the polynomial ring S in n+1 variables, with graded pieces IX(m). For sufficiently large m, depending only on the Hilbert polynomial P of X, all higher cohomology groups of X with coefficients in O(m) vanish, so in particular IX(m) has dimension Q(m) − P(m), where Q is the Hilbert polynomial of projective space.
Pick a sufficiently large value of m. The Q(m) − P(m)-dimensional space IX(m) is a subspace of the Q(m)-dimensional space S(m), so represents a point of the Grassmannian G(Q(m) − P(m), Q(m)). This will give an embedding of the piece of the Hilobert scheme corresponding to the Hilbert polynomial P into this Grassmannian.
It remains to describe the scheme structure on this image, in other words to describe enough elements for the ideal corresponding to it. Enough such elements are given by the conditions that the map from IX(m) ⊗S(k) to S(k+m) has rank at most dim(IX(k+m)) for all positive k, which is equivalent to the vanishing of various determinants. (A more careful analysis shows that it is enough just to take k=1.)
Hilbert schemes can have bad singularities, such as irreducible components that are non-reduced at all points. They can also have irreducible components of unexpectedly high dimension. For example, one might expect the Hilbert scheme of d points (more precisely dimension 0, degree d subschemes) of a scheme of dimension n to have dimension dn, but if n≥3 its irreducible components can have much larger dimension.
"Hilbert scheme" sometimes refers to the punctual Hilbert scheme of 0-dimensional subschemes on a scheme. Informally this can be thought of as something like finite collections of points on a scheme, though this picture can be very misleading when several points coincide.
There is a Hilbert-Chow morphism from the reduced Hilbert scheme of points to the Chow variety of cycles taking any 0-dimensional scheme to its associated 0-cycle. .
The Hilbert scheme of points on is equipped with a natural morphism to an -th symmetric product of . This morphism is birational for M of dimension at most 2. For M of dimension at least 3 the morphism is not birational for large n: the Hilbert scheme is in general reducible and has components of dimension much larger than that of the symmetric product.
The Hilbert scheme of points on a curve C(dimension 1 complex manifold) is isomorphic to a symmetric power of C. It is smooth.
The Hilbert scheme of points on a surface is also smooth (Grothendieck). If , it is a blow-up of a singular subvariety on a symmetric square of . It was used by Mark Haiman in his proof of the positivity of the coefficients of some Macdonald polynomials.
The Hilbert scheme of a smooth manifold of dimension 3 or more is usually not smooth.
Let be a complex Kähler surface with (K3 surface or a torus). The canonical bundle of is trivial, as follows from Kodaira classification of surfaces. hence admits a holomorphic symplectic form. It was observed by Fujiki (for ) and Beauville that is also holomorphically symplectic. This is not very difficult to see, e.g., for . Indeed, is a blow-up of a symmetric square of . Singularities of are locally isomorphic to . The blow-up of is , and this space is symplectic. This is used to show that the symplectic form is naturally extended to the smooth part of the exceptional divisors of . It is extended to the rest of by Hartogs' principle.
A holomorphically symplectic, Kähler manifold is hyperkähler, as follows from Calabi-Yau theorem. Hilbert schemes of points on K3 and a 4-dimensional torus give two series of examples of hyperkähler manifolds: a Hilbert scheme of points on K3 and a generalized Kummer manifold.