CHAPTER 1 Hilbert scheme of points In this chapter, we collect basic facts on the Hilbert scheme of points on a surface. We do not assume the field k is C unless it is explicitly stated. 1.1. General Results on the Hilbert scheme 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 OX(1) an ample line bundle on X. We consider the contravariant functor HilbX from the category of schemes to the category of sets HilbX : [Schemes] [Sets], which is given by HilbX(U) = Z X × U Z is a closed subscheme, Z i X × U π p2 : π is flat⎪⎪ U = U . Namely, HilbX is a functor which associates a scheme U with a set of families of closed subschemes in X parametrized by U. Let π : Z U be the projection. For u U, the Hilbert polynomial in u is defined by Pu(m) = χ(OZ u OX(m)), where Zu = π−1(u). Since Z is flat over U, Pu is independent of u U if U is connected. Conversely, for each polynomial P , let HilbX P be the subfunctor of HilbX which associates U with a set of families of closed subschemes in X parametrized by U which has P as its Hilbert polynomial. Now the basic fact proved by Grothendieck is the following theorem. Theorem 1.1 (Grothendieck [50]). The functor HilbX P is representable by a projective scheme HilbP X . This means that there exists a universal family Z on HilbP X , and that every family on U is induced by a unique morphism φ: U HilbP X . Moreover, if we have an open subscheme Y of X, then we have the correspond- ing open subscheme HilbP Y of HilbP X parametrizing subschemes in Y . In particular, HilbY P is defined for a quasi-projective scheme Y . The proof of this theorem is not given in this book. But we shall give a concrete description when P is a constant polynomial and X is the affine plane A2 (see Theorem 1.9). Once this is established, one can prove the representatibity of a similar functor for a nonsingular complex surface X by a patching argument (see §1.5). In particular, we get HilbP X as a complex manifold, and this is practically 5
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