The Stacks project

Proof. In this proof unadorned products are over $\mathop{\mathrm{Spec}}(k)$. By Proposition 44.3.6 the scheme $H = \underline{\mathrm{Hilb}}^ g_{X/k}$ exists. Consider the universal divisor $D_{univ} \subset H \times X$ and the associated invertible sheaf $\mathcal{O}(D_{univ})$, see Remark 44.3.7. We adjust by tensoring with the pullback via $\sigma _ H : H \to H \times X$ to get

By the Yoneda lemma (Categories, Lemma 4.3.5) the invertible sheaf $\mathcal{L}_ H$ defines a natural transformation

Because $F$ is an open subfuctor, there exists a maximal open $W \subset H$ such that $\mathcal{L}_ H|_{W \times X}$ is in $F(W)$. Of course, this open is nothing else than the open subscheme constructed in Derived Categories of Schemes, Lemma 36.32.3 with $i = 0$ and $r = 1$ for the morphism $H \times X \to H$ and the sheaf $\mathcal{F} = \mathcal{O}(D_{univ})$. Applying the Yoneda lemma again we obtain a commutative diagram

To finish the proof we will show that the top horizontal arrow is an isomorphism.

Let $\mathcal{L} \in F(T) \subset \mathrm{Pic}_{X/k, \sigma }(T)$. Let $\mathcal{N}$ be the invertible $\mathcal{O}_ T$-module such that $Rf_{T, *}\mathcal{L} \cong \mathcal{N}[0]$. The adjunction map

on $X_ T$. Claim: The zero scheme of $s$ is a relative effective Cartier divisor $D$ on $(T \times X)/T$ finite locally free of degree $g$ over $T$.

Let us finish the proof of the lemma admitting the claim. Namely, $D$ defines a morphism $m : T \to H$ such that $D$ is the pullback of $D_{univ}$. Then

Hence $(m \times \text{id}_ X)^*\mathcal{L}_ H$ and $\mathcal{O}(D)$ differ by the pullback of an invertible sheaf on $H$. This in particular shows that $m : T \to H$ factors through the open $W \subset H$ above. Moreover, it follows that these invertible modules define, after adjusting by pullback via $\sigma _ T$ as above, the same element of $\mathrm{Pic}_{X/k, \sigma }(T)$. Chasing diagrams using Yoneda's lemma we see that $m \in h_ W(T)$ maps to $\mathcal{L} \in F(T)$. We omit the verification that the rule $F(T) \to h_ W(T)$, $\mathcal{L} \mapsto m$ defines an inverse of the transformation of functors above.

Proof of the claim. Since $D$ is a locally principal closed subscheme of $T \times X$, it suffices to show that the fibres of $D$ over $T$ are effective Cartier divisors, see Lemma 44.3.1 and Divisors, Lemma 31.18.9. Because taking cohomology of $\mathcal{L}$ commutes with base change (Derived Categories of Schemes, Lemma 36.30.4) we reduce to $T = \mathop{\mathrm{Spec}}(K)$ where $K/k$ is a field extension. Then $\mathcal{L}$ is an invertible sheaf on $X_ K$ with $H^0(X_ K, \mathcal{L}) = K$ and $H^1(X_ K, \mathcal{L}) = 0$. Thus

See Varieties, Definition 33.44.1. To finish the proof we have to show a nonzero section of $\mathcal{L}$ defines an effective Cartier divisor on $X_ K$. This is clear. $\square$