Lemma 107.15.2. The open substack

has the following properties

$\mathcal{C}\! \mathit{urves}^{lci+} \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is smooth,

given a family of curves $X \to S$ the following are equivalent

the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{lci+}$,

$X \to S$ is a local complete intersection morphism and the singular locus of $X \to S$ endowed with any/some closed subspace structure is finite over $S$,

given $X$ a proper scheme over a field $k$ of dimension $\leq 1$ the following are equivalent

the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{lci+}$,

$X$ is a local complete intersection over $k$ and $X \to \mathop{\mathrm{Spec}}(k)$ is smooth except at finitely many points.

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