Given $X$ a k3 surface and $X^{[r]}$ the Hilbert scheme of points on it, I know that the Hilbert-Chow morphism $\rho:X^{[r]}\rightarrow X^{(r)}$ to the symmetric product is birational and, said $D\subset X^{(r)}$ the singular locus, $\rho^{-1}(D)$ is a divisor.

Beauville, in Varietes kahleriennes dont la premiere classe de chern est nulle (page 778), says that given $f:\mathcal{X}\rightarrow B$ a flat family of $X$, i can construct a flat family $g:\mathcal{Y}\rightarrow B$ in which the fibers are $\mathcal{Y}_b=\mathcal{X}_b^{[r]}$. I have some trouble seeing how it's done.

Of course taking the symmetric r-product and the desingularization of each fiber $\mathcal{X}_s$ the dimension of the fibers remains constant (so this checks out with the intuitive idea of flatness). But how can i prove that $g$ is proper and submersive?

  • $\begingroup$ Is your family algebraic? If so, this follows from Grothendieck's original construction of the Hilbert scheme -- which was in the relative setting. If your family is Kaehlerian, but not necessarily algebraic, you should look up "Douady spaces". $\endgroup$ – Jason Starr Jul 1 '14 at 17:03
  • $\begingroup$ can you give me a reference for both cases? $\endgroup$ – Julio Amado Jul 1 '14 at 17:08
  • $\begingroup$ also, what do you mean with "relative setting"? $\endgroup$ – Julio Amado Jul 1 '14 at 17:21
  • $\begingroup$ The reference in the algebraic case is Chapitre IV of the following. MR0146040 (26 #3566) Reviewed. Grothendieck, Alexander. Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957–1962.] (French) Secrétariat mathématique, Paris 1962 ii+205 pp. 14.00 $\endgroup$ – Jason Starr Jul 1 '14 at 17:25
  • $\begingroup$ The reference for the Douady spaces is the following. MR0203082 (34 #2940) Reviewed. Douady, Adrien. Le problème des modules pour les sous-espaces analytiques compacts d'un espace analytique donné. (French) Ann. Inst. Fourier (Grenoble) 16 1966 fasc. 1, 1–95. 32.47 (57.70) $\endgroup$ – Jason Starr Jul 1 '14 at 17:29

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