The Stacks project

Remark 99.15.13. Consider the $2$-fibre product

\[ \xymatrix{ \mathcal{C}\! \mathit{urves}\times _{\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}} \mathcal{P}\! \mathit{olarized}\ar[r] \ar[d] & \mathcal{P}\! \mathit{olarized}\ar[d] \\ \mathcal{C}\! \mathit{urves}\ar[r] & \mathcal{S}\! \mathit{paces}'_{fp, flat, proper} } \]

This fibre product parametrized polarized curves, i.e., families of curves endowed with a relatively ample invertible sheaf. It turns out that the left vertical arrow

\[ \textit{PolarizedCurves} \longrightarrow \mathcal{C}\! \mathit{urves} \]

is algebraic, smooth, and surjective. Namely, this $1$-morphism is algebraic (as base change of the arrow in Lemma 99.14.6), every point is in the image, and there are no obstructions to deforming invertible sheaves on curves (see proof of Lemma 99.15.9). This gives another approach to the algebraicity of $\mathcal{C}\! \mathit{urves}$. Namely, by Lemma 99.15.12 we see that $\textit{PolarizedCurves}$ is an open and closed substack of the algebraic stack $\mathcal{P}\! \mathit{olarized}$ and any stack in groupoids which is the target of a smooth algebraic morphism from an algebraic stack is an algebraic stack.

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