The Stacks project

Proposition 44.6.6. Let $k$ be a separably closed field. Let $X$ be a smooth projective curve over $k$. The Picard functor $\mathrm{Pic}_{X/k}$ is representable.

Proof. Since $k$ is separably closed there exists a $k$-rational point $\sigma $ of $X$, see Varieties, Lemma 33.25.6. As discussed above, it suffices to show that the functor $\mathrm{Pic}_{X/k, \sigma }$ classifying invertible modules trivial along $\sigma $ is representable. To do this we will check conditions (1), (2)(a), (2)(b), and (2)(c) of Lemma 44.5.1.

The functor $\mathrm{Pic}_{X/k, \sigma }$ satisfies the sheaf condition for the fppf topology because it is isomorphic to $\mathrm{Pic}_{X/k}$. It would be more correct to say that we've shown the sheaf condition for $\mathrm{Pic}_{X/k, \sigma }$ in the proof of Lemma 44.4.3 which applies by Lemma 44.6.1. This proves condition (1)

As our subfunctor we use $F$ as defined in Lemma 44.6.2. Condition (2)(b) follows. Condition (2)(a) is Lemma 44.6.4. Condition (2)(c) is Lemma 44.6.5. $\square$

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