Recovering the good component of the Hilbert scheme

Abstract

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

Cited by in corpus (4)

• Gromov-Witten, Gopakumar-Vafa, and Donaldson-Thomas invariants of Calabi-Yau threefolds
• The space of generically étale families
• Moduli of linear representations, symmetric products and the non commutative Hilbert scheme
• Hilbert-Chow morphism for non commutative Hilbert schemes and moduli spaces of linear representations