The Stacks project

Proof. Let $s \in S$. Let $c_0 : X_ s \to Y_0$ be the contraction of Algebraic Curves, Section 53.24 (more precisely Algebraic Curves, Lemma 53.24.2). By Lemma 109.23.3 there exists an elementary étale neighbourhood $(U, u)$ and a morphism $c : X_ U \to Y$ of families of curves over $U$ which recovers $c_0$ as the fibre at $u$. Since $\omega _{Y_0}$ is ample, after possibly shrinking $U$, we see that $Y \to U$ is a stable family of genus $g$ by the openness inherent in Lemmas 109.22.3 and 109.22.5. After possibly shrinking $U$ once more, assertion (1) of the lemma for $c : X_ U \to Y$ follows from Lemma 109.23.1. Moreover, part (2) holds by the uniqueness in Algebraic Curves, Lemma 53.24.2. We conclude that a morphism $c$ as in the lemma exists étale locally on $S$. More precisely, there exists an étale covering $\{ U_ i \to S\} $ and morphisms $c_ i : X_{U_ i} \to Y_ i$ over $U_ i$ where $Y_ i \to U_ i$ is a stable family of curves having properties (1) and (2) stated in the lemma.

To finish the proof it suffices to prove uniqueness of $c : X \to Y$ (up to unique isomorphism). Namely, once this is done, then we obtain isomorphisms

satisfying the cocycle condition (by uniqueness) over $U_ i \times U_ j \times U_ k$. Since $\overline{\mathcal{M}_ g}$ is an algebraic stack, we have effectiveness of descent data and we obtain $Y \to S$. The morphisms $c_ i$ descend to a morphism $c : X \to Y$ over $S$. Finally, properties (1) and (2) for $c$ are immediate from properties (1) and (2) for $c_ i$.

Finally, if $c_1 : X \to Y_ i$, $i = 1, 2$ are two morphisms towards stably families of curves over $S$ satisfying (1) and (2), then we obtain a morphism $Y_1 \to Y_2$ compatible with $c_1$ and $c_2$ at least locally on $S$ by Lemma 109.23.3. We omit the verification that these morphisms are unique (hint: this follows from the fact that the scheme theoretic image of $c_1$ is $Y_1$). Hence these locally given morphisms glue and the proof is complete. $\square$