Hilbert Scheme of Points On A Manifold
"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. (Fogarty 1968, 1969, 1973).
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 (a 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 obtained from by blowing up the diagonal and then dividing by the action induced by . 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.
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