Hilbert scheme

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. 

Remark 1.3.3. As every ideal of colength n in R contains m , we can regard it as an ideal in R/m". Thus the Hilbert scheme Hilb"(fZ/mn) also parametrizes the ideals of colength n in R. We also see that the reduced schemes (Hilbn(ii/m ))re,j are naturally isomorphic for k > n. We will therefore denote these schemes also by Hilb"(R)rei. Hilbn(R)red is the closed subscheme with the reduced induced structure of the Grassmannian Grass(n, R jmn) of n dimensional quotients of R/mn whose geometric points are the ideals of colength n of k[[xi,..., z[Pg.10]

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. ... 

Remark 2.3.12. The Euler numbers of the Hilbert schemes can be expressend in terms of modular forms let q = e2,rlT for r in the upper half plane... 

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... 

Using these results we can also find formulas for the signatures of Hilbert schemes in terms of modular forms. Let again r be in the upper half plane and q = e2mT. Let e and 6 be the following functions ... 

Let X be a smooth projective variety of dimension d over a field k. For d > 3 and n > 4 the Hilbert scheme Xl" is singular. However X 3 is smooth for all d IN. In this section we want to compute the Betti numbers of X can be viewed as a variety of unordered triangles on X. We also consider a number of other varieties of triangles on X, some of which have not yet appeared in the literature. As far as this is not yet known, we show that all these varieties are smooth. We study the relations between these varieties and compute their Betti numbers using the Weil conjectures. 

Now a variety of Ith order data should be a natural smooth compactification of Dlm(X)o- This is for instance the case for D X), as this is given in a canonical way as a subscheme of a product of Hilbert schemes, it is smooth, compact and contains D2m(X)Q as a dense open subvariety. There is a morphism... 

Let be a vector bundle of rank r on a smooth projective variety X. Now we want to study the vector bundles (P) from definition 3.2.6. For this purpose we first consider the bundles Ei on the Hilbert scheme X. We can associate in a natural way to each section sof a section sj of Ei and thus also a section (s)JJ, of... 

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