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
\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
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
\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.
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
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