Let $X$ be a normal, projective, integral variety (over $\mathbb{C}$) and $P$ be the Picard variety parametrizing invertible sheaves on $X$. Does there exist a compactification $\overline{P}$ of $P$ along with an universal family $\mathcal{U}$ defined over $X \times \overline{P}$, flat over $\overline{P}$, parametrizing the space of rank $1$, torsion-free sheaves on $X$ (probably with a fixed Hilbert polynomial)? Any hint/reference will be appreciated.
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4$\begingroup$ For $X$ normal the connected component of ${\rm Pic}_X$ is projective (and over $\mathbf{C}$ it is an abelian variety). This was proved by Grothendieck in FGA, see Kleiman's survey article "The Picard scheme". $\endgroup$– Piotr AchingerCommented Jun 22, 2020 at 20:30
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