Betti number

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. 

Let X be a smooth projective variety over the complex numbers C. Then X is already defined over a finitely generated extension ring R of Z, i.e. there is a variety Xr defined over R such that Xr xr C = X. For every prime ideal p of R let Xp = XR xr R/p. There is a nonempty open subset U C pec(R) such that Xp is smooth for all p U, and the /-adic Betti-numbers of Xp coincide with those of X for all primes / different from the characteristic of A/p (cf. [Kirwan (1) 15.], [Bialynicki-Birula, Sommese (1) 2.]. If m C R is a maximal ideal lying in U for which R/m is a finite field Fq of characteristic p /, we call Xm a good reduction of X modulo q. 

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. ... 

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. 

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... 

The Betti numbers of higher order Kummer varieties... 

In the following two cases we want to compute the Betti numbers of the I Sn- ... 

We will again use the Weil conjectures to determine the Betti numbers of the KSn-. l- To count the points we will use a result from representation theory, the Shintani-descent. Our reference for this is [Digne (1)]. 

In section 2.3 we have obtained power series formulas for the Betti numbers of the S. We now also want to give power series for the KSn-1- They will however not be as nice as those for 5. We define a new multiplication Q on the ring of power series Z[[z,t,w] by... 

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