Let $\mathfrak X \rightarrow S$ a smooth projective family over the spectrum of a dvr. We know that $(\mathfrak X _{\eta_R})^{[2]}$ and $(\mathfrak X _{p})^{[2]}$ are smooth, where $p$ is the closed point of $S$ and $^{[2]}$ stands for Hilbert scheme of two points (or subschemes of lenght $2$). Do we always have for the "global" Hilbert scheme, $\mathfrak X ^{[2]} \rightarrow S$ smooth ?
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2$\begingroup$ Yes, you do have that. The standard obstruction theory for Hilbert schemes (as described in Artin's "Algebraization ... I" or in Chapter 1 of Koll'ar's "Rational curves ...") is actually an obstruction theory relative to a base $S$, at least as long as $\mathfrak{X}$ is flat over $S$ (otherwise there are some corrections). So the usual argument -- length $2$ schemes are LCI and the global obstruction is an $H^1$ that vanishes on zero dimensional schemes -- proves smoothness over $S$ of the relative Hilbert scheme. $\endgroup$– Jason StarrCommented Jul 29, 2015 at 15:12
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$\begingroup$ Thank you for your help. I did not know length 2 subscheme were local complete intersection. Although in Kollar's book there is only the criterion for flatness (theorem I.2.15), it should indeed be easy to conclude from it. $\endgroup$– user3001Commented Jul 30, 2015 at 10:00
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