Lemma 109.4.1 (0DPZ)—The Stacks project

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Part 8: Topics in Moduli Theory
Chapter 109: Moduli of Curves
Section 109.4: The stack of polarized curves
Lemma 109.4.1 (cite)

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Lemma 109.4.1. The morphism $\textit{PolarizedCurves} \to \mathcal{P}\! \mathit{olarized}$ is an open and closed immersion.

Proof. This is true because the $1$-morphism $\mathcal{C}\! \mathit{urves}\to \mathcal{S}\! \mathit{paces}'_{fp, flat, proper}$ is representable by open and closed immersions, see Quot, Lemma 99.15.12. $\square$

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