The Stacks project

Remark 44.3.7. Let $X$ be a geometrically irreducible smooth proper curve over a field $k$ as in Proposition 44.3.6. Let $d \geq 0$. The universal closed object is a relatively effective divisor

over $\underline{\mathrm{Hilb}}^{d + 1}_{X/k}$ by Lemma 44.3.1. In fact, $D_{univ}$ is isomorphic as a scheme to $\underline{\mathrm{Hilb}}^ d_{X/k} \times _ k X$, see proof of Lemma 44.3.4. In particular, $D_{univ}$ is an effective Cartier divisor and we obtain an invertible module $\mathcal{O}(D_{univ})$. If $[D] \in \underline{\mathrm{Hilb}}^{d + 1}_{X/k}$ denotes the $k$-rational point corresponding to the effective Cartier divisor $D \subset X$ of degree $d + 1$, then the restriction of $\mathcal{O}(D_{univ})$ to the fibre $[D] \times X$ is $\mathcal{O}_ X(D)$.