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Let $H$ be an irreducible component of a Hilbert scheme of surfaces in $\mathbb{P}^n_{\mathbb{C}}$ whose general point corresponds to a smooth irreducible surface.

Consider the function $\chi:U\to\mathbb{Z}$ defined over the subset $U\subseteq H$ parametrizing smooth irreducible surfaces which sends a surface $S$ to its topological Euler-Poincaré characteristic $\chi(S)$.

What property does the function $\chi$ have? For example, is it constant?

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    $\begingroup$ Because $H$ is irreducible and $U$ is open, $U$ is connected, so yes, $\chi$ is constant. This has nothing to do with irreducibility or two-dimensionality of the surfaces - it's a general topological fact about smooth proper maps. $\endgroup$ – Will Sawin Feb 3 at 13:00
  • $\begingroup$ Thanks for the quick answer. Can you give me a reference for the fact that $\chi$ is a smooth proper map? $\endgroup$ – gio Feb 3 at 13:11
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    $\begingroup$ If you're fine reducing to the analytic setting, en.wikipedia.org/wiki/Ehresmann%27s_lemma $\endgroup$ – Will Sawin Feb 3 at 13:26

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