The Stacks project

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

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

We denote this $2$-fibre product by

\[ \textit{PolarizedCurves} = \mathcal{C}\! \mathit{urves}\times _{\mathcal{S}\! \mathit{paces}'_{fp, flat, proper}} \mathcal{P}\! \mathit{olarized} \]

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

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

  2. $\mathcal{L}$ 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

\[ \xymatrix{ X' \ar[d] \ar[r]_ f & X \ar[d] \\ S' \ar[r]^ g & S } \]

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.

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. $\square$

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

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

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 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). $\square$

Comments (4)

Comment #2520 by on

There seems to be a missing dollar sign somewhere in the LaTeX that is causing some problems. In the last sentence of the first pargraph of the proof of 0DQ0, "amps" should be "maps".

Comment #2521 by on

It's not quite a missing dollar in the TeX code (Johan would notice this when building the pdf) but an issue with the parser. I'll try to fix this as soon as possible. Thanks!

Comment #2522 by on

@PieterBelmans. You are right. The PDF file is fine. Still, "amps" should be "maps".

Comment #2527 by on

Thanks Jason! I changed amps to maps and I fiddled with the LaTeX untill the page looked ok.

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