The Stacks project

Lemma 107.4.3. Let $X \to S$ be a family of curves. Then there exists an ├ętale 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 107.4.2. 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 ├ętale covering $\{ S_ i \to S\} $, see More on Morphisms, Lemma 37.34.7. After replacing $S_ i$ by a suitable open covering we may assume $X_ i \to S_ i$ is H-projective, see Morphisms, Lemmas 29.42.6 and 29.42.4 (this is also discussed in detail in More on Morphisms, Section 37.45). $\square$


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