Definition 109.20.1 (0E6T)—The Stacks project

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

$f$ is at-worst-nodal of relative dimension $1$ , and
$f_*\mathcal{O}_ X = \mathcal{O}_ S$ and this holds after any base change ¹.

[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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