Let $X$ be a smooth projective threefold. Let $I_n(X,\beta)$ be the Hilbert scheme parametrizing subschemes $Z \subset X$ with curve class $\beta \in H_2(X,\mathbb{Z})$ and $\chi(\mathcal{O}_Z)=n$. Can one understand the second condition $\chi(\mathcal{O}_Z)=n$ geometrically?

I am aware that when $\beta=0$, $I_n(X,0)=X^{[n]}$ is the Hilbert schemes of $n$ points on $X$. How should one understand $\chi(\mathcal{O}_Z)=n$ for general $\beta$?