Recovering the good component of the Hilbert scheme
Abstract
Description
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.
Several sections are rewritten, and results are now in a more general context. In particular flatness, finite type and noetherian assumptions are removed from the main result. Inaccuracies and mistakes have been corrected
Several sections are rewritten, and results are now in a more general context. In particular flatness, finite type and noetherian assumptions are removed from the main result. Inaccuracies and mistakes have been corrected