Lemma 107.4.2. The morphism $\textit{PolarizedCurves} \to \mathcal{C}\! \mathit{urves}$ is smooth and surjective.

Proof. Surjective. Given a field $k$ and a proper algebraic space $X$ over $k$ of dimension $\leq 1$, i.e., an object of $\mathcal{C}\! \mathit{urves}$ over $k$. By Spaces over Fields, Lemma 70.9.3 the algebraic space $X$ is a scheme. Hence $X$ is a proper scheme of dimension $\leq 1$ over $k$. By Varieties, Lemma 33.42.4 we see that $X$ is H-projective over $\kappa$. In particular, there exists an ample invertible $\mathcal{O}_ X$-module $\mathcal{L}$ on $X$. Then $(X, \mathcal{L})$ is an object of $\textit{PolarizedCurves}$ over $k$ which maps to $X$.

Smooth. Let $X \to S$ be an object of $\mathcal{C}\! \mathit{urves}$, i.e., a morphism $S \to \mathcal{C}\! \mathit{urves}$. It is clear that

$\textit{PolarizedCurves} \times _{\mathcal{C}\! \mathit{urves}} S \subset \mathcal{P}\! \mathit{ic}_{X/S}$

is the substack of objects $(T/S, \mathcal{L}/X_ T)$ such that $\mathcal{L}$ is ample on $X_ T/T$. This is an open substack by Descent on Spaces, Lemma 72.12.2. Since $\mathcal{P}\! \mathit{ic}_{X/S} \to S$ is smooth by Moduli Stacks, Lemma 106.8.5 we win. $\square$

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