Hilbert Schemes of Points on Surfaces

Abstract

n distinct points of X, if d ≥ 2 (Lemma 7.1.6). On the other hand, every X (n) is nonsingular if d = 1 (Exercise 7.1.E.5).Section 7.2 presents some general results on Hilbert schemes of points on quasiprojective schemes: their existence (Theorem 7.2.3) and the description of their Zariski tangent spaces (Lemma 7.2.5). Also, the punctual Hilbert scheme X [n] x (parametrizing length-n subschemes supported at a given point x) is introduced, and it is shown that X [n] x is projective and connected (Proposition 7.2.9). The Hilbert-Chow morphism is introduced and studied in Section 7.3, again in the setting of quasi-projective schemes. The existence of a projective morphism of schemes γ : X [n] −→ X (n) which yields the cycle map on closed points is deduced from work of Iversen (Theorem 7.3.1). The fibers of γ are products of punctual Hilbert schemes; their connectedness implies that X [n] is connected if X is (Corollary 7.3.4). Then, the loci X [n] * * , resp. X [n] * , consisting of subschemes supported at n distinct points, resp. at least n − 1 distinct points, are considered. In particular, it is shown that X [n] * is a nonsingular variety of dimension nd, and the complement X [n] * \ X [n] * * is a nonsingular 208 Chapter 7. Hilbert Schemes of Points prime divisor, if X is a nonsingular variety of dimension d (Lemma 7.3.5). Section 7.4 is devoted to Hilbert schemes of points on a nonsingular surface X. Each X [n] is shown to be a nonsingular variety of dimension 2n (Theorem 7.4.1) and the Hilbert-Chow morphism is shown to be birational, with exceptional set being a prime divisor (Proposition 7.4.5). Finally, the Hilbert-Chow morphism is shown to be crepant (Theorem 7.4.6), a result due to Beauville in characteristic 0 and to Kumar-Thomsen in characteristic p ≥ 3.Section 7.5 begins with the observation that any symmetric product of a split quasiprojective scheme is split as well (Lemma 7.5.1). Together with the crepantness of the Hilbert-Chow morphism, this implies the splitting of X [n] , where X is a nonsingular split surface (Theorem 7.5.2). In turn, this yields the vanishing of higher cohomology groups of any ample invertible sheaf on X [n] , if X is split and proper over an affine variety (Corollary 7.5.4). This applies, in particular, to the nonsingular projective split surfaces and also to the nonsingular affine surfaces (since these are split by Proposition 1.1.6). Further, we obtain a relative vanishing result for the Hilbert-Chow morphism (Corollary 7.5.5).Notation. Throughout this chapter, X denotes a quasi-projective scheme over an algebraically closed field k of characteristic p ≥ 0, and n denotes a positive integer. By schemes, as earlier in the book, we mean Noetherian separated schemes over k; their closed points will just be called points.