Recovering the Good Component of the Hilbert Scheme

We give an explicit construction, for a flat map X → S of algebraic spaces, of an ideal in the n’th symmetric product of X over S. Blowing up this ideal is then shown to be isomorphic to the schematic closure in the Hilbert scheme of length n subschemes of the locus of n distinct points. This generalises Haiman’s corresponding result ([13]) for the affine complex plane. However, our construction of the ideal is very different from that of Haiman, using the formalism of divided powers rather than representation theory. In the non-flat case we obtain a similar result by replacing the n’th symmetric product by the n’th divided power product. The Hilbert scheme, HilbnX/S , of length n subschemes of a scheme X over some S is in general not smooth even if X → S itself is smooth. Even worse, it may not even be (relatively) irreducible. In the case of the affine plane over the complex numbers (where the Hilbert scheme is smooth and irreducible) Haiman (cf., [13]) realised the Hilbert scheme as the blow-up of a very specific ideal of the n’th symmetric product of the affine plane. It is the purpose of this article to generalise Haiman’s construction. As the Hilbert scheme in general is not irreducible while the symmetric product is (for a smooth geometrically irreducible scheme over a field say) it does not seem reasonable to hope to obtain a Haiman like description of all of HilbnX/S and indeed we will only get a description of the schematic closure of the open subscheme of n distinct points. With this modification we get a general result which seems very close to that of Haiman. The main difference from the arguments of Haiman is that we need to define the ideal that we want to blow up in a general situation and Haiman’s construction seems to be too closely tied to the 2-dimensional affine space in characteristic zero. As a bonus we get that our constructions work very generally. We have thus tried to present our results in a generality that should cover reasonable applications (encouragement from the referee has made us make it more general than we did in a previous version of this article). There are some rather immediate consequences of this generality. The first one is that we have to work with algebraic spaces instead 2000 Mathematics Subject Classification. 14C05.