The Hilbert scheme

Let T be a locally noetherian scheme, X a quasiprojective scheme over T and C a very ample invertible sheaf on X over T. 

Definition 1.1.1. [Grothendieck (1)] Let HUb(X/T) be the contravariant functor from the category SchlnT of locally noetherian T-schemes to the category Ens of sets, which for locally noetherian T-schemes U,V and a morphism j V — U is given by 

Theorem 1.1.2 [Grothendieck (1)]. Let X be projective over T. Then for every polynomial P Q[ ] the functor Ttilbp(X/T) is representable by a projective T-scheme Hilbp(X/T). Hilb X/T) is represented by 

For an open subscheme Y C X the functor Hilbp(Y/T) is represented by an open subscheme 

Definition 1.1.3. Hilb (X/T) is the Hilbert scheme of X over T. If T is spec(k) for a field k, we will write Hilb(X) instead of Hilb(X/T) and Hilbp(X) instead of Hilbp(X/T). If P is the constant polynomial P = n, then Hilbn(X/T) is the relative Hilbert scheme of subschemes of length n on X over T. If T is the spectrum of a field, we will write X for Hilb (X) = Hilbn(X/spec(fc)). Xtn) is the Hilbert scheme of subschemes of length n on X. 

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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]

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... 

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. 

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... 

The Hodge numbers of the Hilbert scheme of points on a smooth projective surface, preprint 1993. 

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

Towards the Chow ring of the Hilbert scheme of P2, J. reine angew. Math. 441 (1993), 33-44. 

Connectedness of the Hilbert scheme, Publ. Math. IHES 29 (1966), 261-304. 

Multiple point formulas II the Hilbert scheme, Enumerative Geometry, Proc Sitjes 1987, S. Xambo-Descamps, ed., Lecture Notes in Math. 1436, Springer-Verlag, Berlin Heidelberg 1990, 101-138. 

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. 

Moduli spaces parameterizing objects associated with a given space X are rich source of spaces with interesting structures. They usually inherits structures of X, but sometimes even more they have more structures than X, or pull out hidden structures of X. The purpose of this note is to add an example of these phenomena. We study the moduli space parameterizing 0-dimensional subschemes of length u in a nonsingular quasi-projective surface X over C. It is called the Hilbert scheme of points, and denoted by X ... 

However, when X is 1-dimensional, we have unique 1-dimensional subspace in T X. In fact, the Hilbert scheme Xt" is isomorphic to S X when dimX = 1. 

These study will lead us to wonder the reason why we encounter objects, such as modular forms, affine Lie algebras, the conformal held theory etc. Usually we think these objects live with elliptic curves, not with surfaces. Formally, we have two possibilities one is that these objects are so universal (like Dynkin diagrams) that they appear everywhere. The other possiblity is that elliptic curves are hidden in the Hilbert schemes. We do not know which is correct at this moment, but we believe that the second one is correct. 

In fact, we think the space which is really relevant is the generating function of the Hilbert schemes ... 

First, we recall the definition of the Hilbert scheme in general (not necessarily of points, nor on a surface). Let X be a projective scheme over an algebraically closed field k and Gx(l) an ample line bundle on X. We consider the contra,variant functor Hilbx from the... 

For example, the Hilbert scheme of n points in the affine line A is... 

The purpose of this chapter is to construct a hyper-Kahler metric on the Hilbert scheme of n points on C. This will be accomplished by identifying with a hyper-... 

Let us construct its resolution using the Hilbert scheme of points on as follows. Consider the Hilbert scheme where N is the order of P. The P-action on... 

Question 4.6 (Hitchin). Gonsider a hnite group action on a K3 surface which preserves a hyper-Kahler structure. (Such actions were classified by Mukai [58].) It naturally induces the action on the Hilbert scheme of points on the K3 surface. Its fixed point component is a compact hyper-Kahler manifold as in 4.2. Is the component a new hyper-Kahler manifold The known compact irreducible hyper-Kahler manifolds are equivalent to the Hilbert scheme of points on a /F3 surface, or the higher order Kummar variety (denoted by Kr in [6]) modulo deformation and birational modification, (cf. [57, p.l68j)... 

This is the Kronheimer s construction. Since the hyper-Kahler manifolds which are obtained as the F-fixed component of the Hilbert scheme (or its deformation) satisfy (o =... 

Question 4.12. Is it possible to give an analogue of the McKay correspondence between irreducible representations of the symmetric group and cycles in the Hilbert scheme (C ) " It seems likely that the tautological bundle V plays a fundamental role. 

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