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Recovering the good component of the Hilbert scheme. (English) Zbl 1300.14004

The Hilbert scheme \({\text{Hilb}}^n_{X/S}\) parametrizing length \(n\) closed subschemes of \(X\) over \(S\) continues to draw great interest from algebraic geometers. The case \(X = \mathbb P^N\) is already interesting, as \({\text{Hilb}}^n_{\mathbb P^N/k}\) is smooth and irreducible for \(N=2\), but reducible for \(N = 3\) and \(n\) large [A. Iarrobino, Invent. Math. 15, 72–77 (1972; Zbl 0227.14006)] and hence singular by R. Hartshorne’s connectedness theorem [Publ. Math. Inst. Hautes Études Sci. 29, 5–48 (1966; Zbl 0171.41502)]. Motivated by Haiman’s construction of \({\text{Hilb}}^n_{\mathbb A^2/\mathbb C}\) as the blow-up of \(\text{Sym}^n (\mathbb A^2)\) at a concrete ideal [Discrete Math. 193, No. 1–3, 201–224 (1998; Zbl 1061.05509)], the authors construct the good component \(G^n_{X/S} \subset {\text{Hilb}}^n_{X/S}\), the closure of subschemes consisting of \(n\) distinct points, as a concrete blow-up of a symmetric product for separated morphisms \(f:X \to S\) of algebraic spaces.
Working at this level of generality, the authors need to show existence of the Hilbert scheme as an algebraic space, extending M. Artin’s result [in: Global Analysis, Papers in Honor of K. Kodaira 21–71 (1969; Zbl 0205.50402)] for \(f\) locally of finite presentation. In this context, the \(n\)th symmetric product need not exist, so instead they use the \(n\)th divided power product \(\Gamma^n_{X/S}\) (the affine model is due to N. Roby [C. R. Acad. Sci., Paris, Sér. A 290, 869–871 (1980; Zbl 0471.13008)]), which is homeomorphic to \(\text{Sym}^n X\) in general and isomorphic when \(f\) is flat [D. Rydh, “Families of zero cycles and divided powers: I. Representability”, arXiv:0803.0618]. When \(X\) and \(S\) are affine, the authors define the ideal of norms \(I\) in the ring corresponding to \(\Gamma^n_{X/S}\) and show that these patch together to define a closed subscheme \(\Delta_X \subset \Gamma^n_{X/S}\). With this machinery in place, the authors prove that if \(f: X \to S\) is a separated morphism of algebraic spaces, then \(G^n_{X/S} \subset {\text{Hilb}}^n_{X/S}\) is isomorphic to the blow-up of \(\Gamma^n_{X/S}\) along the closed subspace \(\Delta_X\). In the important case that \(f\) is flat, \(G^n_{X/S}\) is obtained by blowing up the geometric quotient \(X^n_S/S_n\). As a byproduct of their method, they show that \(G^n_{X/S} = {\text{Hilb}}^n_{X/S}\) is smooth for \(f\) smooth and separated of relative dimension two, extending the result of J. Fogarty [Am. J. Math. 90, 511–521 (1968; Zbl 0176.18401)].

MSC:

14C05 Parametrization (Chow and Hilbert schemes)
14A20 Generalizations (algebraic spaces, stacks)
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References:

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