# Euler-Poincaré characteristic as a function from a Hilbert scheme of surfaces

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?

• 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. – Will Sawin Feb 3 at 13:00
• Thanks for the quick answer. Can you give me a reference for the fact that $\chi$ is a smooth proper map? – gio Feb 3 at 13:11
• If you're fine reducing to the analytic setting, en.wikipedia.org/wiki/Ehresmann%27s_lemma – Will Sawin Feb 3 at 13:26