Lemma 44.6.4. Let $k$ be a field. Let $X$ be a smooth projective curve of genus $g$ over $k$ with a $k$-rational point $\sigma$. The open subfunctor $F$ defined in Lemma 44.6.2 is representable by an open subscheme of $\underline{\mathrm{Hilb}}^ g_{X/k}$.

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

$\mathcal{L}_ H = \mathcal{O}(D_{univ}) \otimes _{\mathcal{O}_{H \times X}} \text{pr}_ H^*\sigma _ H^*\mathcal{O}(D_{univ})^{\otimes -1} \in \mathrm{Pic}_{X/k, \sigma }(H)$

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

$h_ H \longrightarrow \mathrm{Pic}_{X/k, \sigma }$

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

$\xymatrix{ h_ W \ar[d] \ar[r] & F \ar[d] \\ h_ H \ar[r] & \mathrm{Pic}_{X/k, \sigma } }$

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 $Rf_{T, *}\mathcal{L} \cong \mathcal{N}$. The adjunction map

$f_ T^*\mathcal{N} \longrightarrow \mathcal{L} \quad \text{corresponds to a section }s\text{ of}\quad \mathcal{L} \otimes f_ T^*\mathcal{N}^{\otimes -1}$

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

$(m \times \text{id}_ X)^*\mathcal{O}(D_{univ}) \cong \mathcal{O}_{T \times X}(D)$

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

$\deg (\mathcal{L}) = \chi (X_ K, \mathcal{L}) - \chi (X_ K, \mathcal{O}_{X_ K}) = 1 - (1 - g) = g$

See Varieties, Definition 33.43.1. 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. $\square$

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