Recovering the good component of the Hilbert scheme

Details Recovering the good component of the Hilbert scheme Recovering the good component of the Hilbert scheme

2004-05-05

arxiv.org/abs/math/0405073v2

Mathematics

Algebraic Geometry

Several sections are rewritten, and results are now in a more general context. In particular flatness, finite type and noether

Scientific paper

In the Hilbert scheme of points on a scheme X there is an open subset parameterizing distinct points. The closure of that open set is by definition the good component. When X is flat over the base, we show that a certain blow-up of the symmetric product of X is the good component. The center of the blow-up we describe by giving generators for its defining ideal. In the non-flat case we obtain similar result by replacing the symmetric product with the divided power product. For smooth surfaces X the good component equals the Hilbert scheme of points.

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