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

## 109.4 The stack of polarized curves

In this section we work out some of the material discussed in Quot, Remark 99.15.13. Consider the $2$-fibre product

We denote this $2$-fibre product by

This fibre product parametrizes polarized curves, i.e., families of curves endowed with a relatively ample invertible sheaf. More precisely, an object of $\textit{PolarizedCurves}$ is a pair $(X \to S, \mathcal{L})$ where

$X \to S$ is a morphism of schemes which is proper, flat, of finite presentation, and has relative dimension $\leq 1$, and

$\mathcal{L}$ is an invertible $\mathcal{O}_ X$-module which is relatively ample on $X/S$.

A morphism $(X' \to S', \mathcal{L}') \to (X \to S, \mathcal{L})$ between objects of $\textit{PolarizedCurves}$ is given by a triple $(f, g, \varphi )$ where $f : X' \to X$ and $g : S' \to S$ are morphisms of schemes which fit into a commutative diagram

inducing an isomorphism $X' \to S' \times _ S X$, in other words, the diagram is cartesian, and $\varphi : f^*\mathcal{L} \to \mathcal{L}'$ is an isomorphism. Composition is defined in the obvious manner.

**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.
$\square$

Lemma 109.4.2. 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 72.9.3 the algebraic space $X$ is a scheme. Hence $X$ is a proper scheme of dimension $\leq 1$ over $k$. By Varieties, Lemma 33.43.4 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

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 74.13.2. Since $\mathcal{P}\! \mathit{ic}_{X/S} \to S$ is smooth by Moduli Stacks, Lemma 108.8.5 we win. $\square$

Lemma 109.4.3. Let $X \to S$ be a family of curves. Then there exists an étale covering $\{ S_ i \to S\} $ such that $X_ i = X \times _ S S_ i$ is a scheme. We may even assume $X_ i$ is H-projective over $S_ i$.

**Proof.**
This is an immediate corollary of Lemma 109.4.2. Namely, unwinding the definitions, this lemma gives there is a surjective smooth morphism $S' \to S$ such that $X' = X \times _ S S'$ comes endowed with an invertible $\mathcal{O}_{X'}$-module $\mathcal{L}'$ which is ample on $X'/S'$. Then we can refine the smooth covering $\{ S' \to S\} $ by an étale covering $\{ S_ i \to S\} $, see More on Morphisms, Lemma 37.38.7. After replacing $S_ i$ by a suitable open covering we may assume $X_ i \to S_ i$ is H-projective, see Morphisms, Lemmas 29.43.6 and 29.43.4 (this is also discussed in detail in More on Morphisms, Section 37.50).
$\square$

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