Cell decomposition

In section 2.2 we consider the punctual Hilbert schemes Hilbn(fc[[x, y]]). We give a cell decomposition of the strata and so determine their Betti numbers. I have published most of the results of this section in a different form in [Gottsche (3)]. They have afterwards been used in [Iarrobino-Yameogo (1)] to study the structure of the cohomology ring of the Gt- We also recall the results of [Ellingsrud-Str0mme (1),(2)] on a cell decomposition of Hilb"(fc[[x,j/]]) and P. ... 

Let k be an algebraically closed field. In this section we review the methods of [Ellingsrud-Str0mme(l)] for the determination of a cell decomposition of and modify them in order get a cell decomposition and thus (for k = C) the homology of the strata Zt and Gt of Hilb"(fc[[x, y]]). Let R = fc[[ ,y]]. Let Hilb (A2,0) be the closed subscheme with the induced reduced structure of (A2) parametrizing subschemes with support 0. By lemma 2.1.4 we have... 

Ellingsrud and Strpmme have constructed the cell decomposition of P using the following results of [Bialynicki-Birula (1),(2)]. Let X be a smooth projective variety over k with an action of the multiplicative group Gm. We will denote this action by Let x X be a fixed point of this action. Let T x C Tx,x be the linear subspace on which all the weights of the induced action of Gm are positive. 

L C P2 the line T2 = 0, F = L Po, P2 = P2 L. Then induces the cell decomposition of P2 into Po, Pi, P2. Ellingsrud and Strpmme apply theorem 2.2.3 to the induced Gm-action on P - We will modify their arguments in order to obtain a cell decomposition of the strata Zt of Hilbn(P). 

We denote by the action of Gm on P induced by d>. As it has only finitely many fixed points, it gives a cell decomposition of P - Hilbn(P)re(i = Hilbn(A2,0) C P is the subvariety parametrizing subschemes Z of colength n with support supp(Z) = Po. If Z P has support Po, then... 

So by theorem 2.2.3 Hilb (A2,0) is a union of cells of the cell decomposition of P which belong to fixed points in Hilbn(A2,0). In particular Hilbn(A2,0) has a cell decomposition. 

We now formulate our result on the cell decompositions of Zt and Gt in a form which has been influenced by [Iarrobino-Yameogo (1)]. In particular the formula for the Betti numbers of Gt does not follow immediately from my original formulation. In [Iarrobino-Yameogo (1)] two combinatorical formulas axe shown in order to derive this formula from my original one in [Gottsche (4)]. Here we will give a direct proof. 

We consider the induced Gro-action on Hilb"(fi). We know already that it gives a cell decomposition of Hilb (JJ). Let T = (tt) be a sequence of non-negative integers with T = n. 

The Gm-action induces a cell decomposition of Gt- Its cells are the intersections of the cells of Zt with Gt-... 

As P has a cell decomposition, we get (2) from the theorem of Leray-Hirsch for Chow groups [Collino-Fulton (1)] and 4.4.2. ... 

On a cell decomposition of the Hilbert scheme of points in the plane, Invent. Math 91 (1988), 365-370. 

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