Lemma 109.15.2 (0DZX)—The Stacks project

Lemma 109.15.2. The open substack

\[ \mathcal{C}\! \mathit{urves}^{lci+} = \mathcal{C}\! \mathit{urves}^{lci} \cap \mathcal{C}\! \mathit{urves}^{+} \subset \mathcal{C}\! \mathit{urves} \]

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
1. the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{lci+}$,
2. $X \to S$ 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
1. the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{lci+}$,
2. $X$ is a local complete intersection over $k$ and $X \to \mathop{\mathrm{Spec}}(k)$ is smooth except at finitely many points.

Proof. If we can show that there is an open substack $\mathcal{C}\! \mathit{urves}^{lci+}$ whose points are characterized by (2), then we see that (1) holds by combining Lemma 109.15.1 with Deformation Problems, Lemma 93.16.4. Since

\[ \mathcal{C}\! \mathit{urves}^{lci+} = \mathcal{C}\! \mathit{urves}^{lci} \cap \mathcal{C}\! \mathit{urves}^{+} \]

inside $\mathcal{C}\! \mathit{urves}$, we conclude by Lemmas 109.13.1 and 109.14.1. $\square$

The Stacks project

Comments (0)

Post a comment