Betti numbers of Hilbert schemes

The second chapter is devoted to computing the Betti numbers of Hilbert schemes of points. The main tool we want to use are the Weil conjectures. In section 2.1 we will study the structure of the closed subscheme of X which parametrizes subschemes of length nonl concentrated in a variable point of X. We will show that (X )rei is a locally trivial fibre bundle over X in the Zariski topology with fibre Hilbn( [[xi. arj]]). We will then also globalize the stratification of Hilbn( [[xi. x ]]) from section 1.3 to a stratification of Some of the strata parametrize higher order data of smooth m-dimensional subvarieties Y C X for m d. In chapter 3 we will study natural smooth compactifications of these strata. 

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.  

In section 2.5 we study varieties of triangles. As mentioned above is smooth for an arbitrary smooth projective variety X. So we can use the Weil 

The variety Hilb (X) of triangles on X with a marked side has been used in 

The second chapter is devoted to computing the Betti numbers of Hilbert schemes of points. The main tool we want to use are the Weil conjectures. In section 2.1 we will study the structure of the closed subscheme of which parametrizes subschemes of length n on X concentrated in a variable point of X. We will show that is a locally trivial fibre bundle over X in the Zariski topology 

Let k be a (not necessarily algebraically closed) field and X a smooth quasipro-jective variety of dimension d over k. In this section we study the structure of the stratum ()red which parametrizes subschemes of X which are concentrated in a (variable) point in X. 

Proof Let k be an algebraic closure of k and X = X x k. Let Z C X be a subscheme of length n of X concentrated in a point, Iz its ideal in the local ring Ox,z and mx,z the maximal ideal of Ox,x Then we have Iz D x (cf. 1.3.2). So we see that Hilbn(An/X)re j and (Xj ) red are closed subschemes of Xf l with the reduced induced structure, which have the same geometric points. Thus they are equal. The assertion on 7r follows directly from the definitions.  

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We will use the Weil conjectures to compute the Betti numbers of Hilbert schemes. They have been used before to compute Betti numbers of algebraic varieties, at least since in [Harder-Narasimhan (1)] they were applied for moduli spaces of vector bundles on smooth curves. 

One would expect that similar formulas as for the Betti numbers of Hilbert schemes of points also hold for their Hodge numbers. For a smooth projective variety X over C let W (X) = dimH (X,9.px) be the p,q)th Hodge number and let... 

In chapter 1 we recall some fundamental facts, that will be used in the rest of the book. First in section 1.1, we give the definition and the most important properties of Xfnl then in section 1.2 we explain the Weil conjectures in the form in which we are later going to use them in order to compute Betti numbers of Hilbert schemes, and finally in section 1.3 we introduce the punctual Hilbert scheme, which parametrizes subschemes concentrated in a point of a smooth variety. We hope that the non-expert reader will find in particular sections 1.1 and 1.2 useful as a quick reference. 

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