The Weil conjectures

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 projective scheme over a finite field Fq, let Fq be an algebraic closure of Fq and X = X x IFq. The geometric Frobenius 

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. 

If X and S are good reductions of smooth varieties Y and U over C, we have  

Then there axe rational numbers ni.nr such that 

Then there are rational numbers ni. nr such that 

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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. 

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)]. 

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. 

To compute the Betti numbers of some of these varieties we will again use the Weil conjectures. So we have to count their points over finite fields. First we look at the local situation. Let k be a field and R =. .., x,j]]. As above Hilb"(ii)... 

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. 

KAn-i for A an abelian surface, using the Weil conjectures. Here KAn-i is a symplectic manifold, defined as the kernel of the map —> A given by composing... 

The Weil conjectures are a powerful tool whose use is not as widely spread as it could be we hope that the applications given in chapter 2 will convince the reader that they are not only important theoretically, but also quite useful in many concrete cases. 

To read this book the reader only needs to know the basics of algebraic geometry. For instance the knowledge of [Hartshorne (1)], is certainly enough, but also that of [Eisenbud-Harris (1)] suffices for reading most parts of the book. At some points a certain familiarity with the functor of points (like in the last chapter of [Eisenbud-Harris (1)]) will be useful. Of course we expect the reader to accept some results without proof, like the existence of the Hilbert scheme and obviously the Weil conjectures. 

In this chapter we shall calculate the Poincare polynomial of (C2). This was first accomplished by Ellingsrud and Strpmme [14]. They have used the Bialynicki-Birula decomposition associated with the natural torus action on (C2), and then compute the Poincare polynomial using the Weil conjecture. Our approach is essentially the same, but we use Morse theory instead of the Weil conjecture. 

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... 

We want to use the Weil conjectures to compute the Betti numbers of for a smooth projective surface S over C. Let X be a smooth projective variety of dimension d over a field k. Let R -, ar, ]]. We denote Vn = Hilb l(i )re[Pg.29]

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