## On the place where $\mathrm{Hilb}_{lines}^{x}(X)$ is smooth.

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Let `$X\subset \mathbb{P}_{\mathbb{C}}^N$` be irreducible generically smooth closed subscheme and let `$\mathrm{Hilb}_{lines}^{x}(X)$` denote the Hilbert scheme of lines contained in $X$ and passing through the point $x\in X$. Is it true that the set ```$$
\{ x\in X : \mathrm{Hilb}_{lines}^{x}(X) \mbox{ is smooth } \}
$$``` is constructible?

Thanks in advance.

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## 1 Answer

Yes. This can be seen as follows:

Let $Hilb_{lines}(X)$ denote the Hilbert scheme of lines in $X$ and let $\Gamma \subset X \times Hilb_{lines}(X)$ be the correponding universal family. Then $Hilb_{lines}^x(X) = p^{-1}(x)$ where $p:\Gamma \to X$ is induced by the first projection. So we are reduced to the folowing (well-known) statement:

Let $f:Y \to X$ be a proper morphism of finite type schemes over a field. Then the set `$\{x \in X | f^{-1}(x) \mbox{ is smooth } \}$` is constructible.

To prove this, by replacing $X$ by $X_{red}$ and $Y$ by `$Y \times_X X_{red}$` we may assume $X$ is reduced (since we only care about the fibres). By generic flatness, we may find a finite stratification of $X$ by locally closed reduced subschemes $X_i$ so that the induced morphisms $Y \times_X X_i \to X_i$ are all flat. For a flat proper morphism the locus of points in the base so that the fibres are smooth is open. It follows that the set we are interested in is a finite union of open subsets of closed subsets of $X$, so is constructible.

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Thank you very much. – gio Jun 28 2011 at 17:07
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