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?