The Stacks project

Definition 109.20.1. Let $f : X \to S$ be a family of curves. We say $f$ is a prestable family of curves if

  1. $f$ is at-worst-nodal of relative dimension $1$, and

  2. $f_*\mathcal{O}_ X = \mathcal{O}_ S$ and this holds after any base change1.

[1] In fact, it suffices to require $f_*\mathcal{O}_ X = \mathcal{O}_ S$ because the Stein factorization of $f$ is étale in this case, see More on Morphisms of Spaces, Lemma 76.36.9. The condition may also be replaced by asking the geometric fibres to be connected, see Lemma 109.11.2.

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