Lemma 109.18.1 (0DSY)—The Stacks project

Table of contents
Part 8: Topics in Moduli Theory
Chapter 109: Moduli of Curves
Section 109.18: Nodal curves
Lemma 109.18.1 (cite)

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Lemma 109.18.1. There exist an open substack $\mathcal{C}\! \mathit{urves}^{nodal} \subset \mathcal{C}\! \mathit{urves}$ such that

given a family of curves $f : X \to S$ the following are equivalent
1. the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{nodal}$,
2. $f$ is at-worst-nodal of relative dimension $1$,
given $X$ a scheme proper over a field $k$ with $\dim (X) \leq 1$ the following are equivalent
1. the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{nodal}$,
2. the singularities of $X$ are at-worst-nodal and $X$ is equidimensional of dimension $1$.

Proof. In fact, it suffices to show that given a family of curves $f : X \to S$, there is an open subscheme $S' \subset S$ such that $S' \times _ S X \to S'$ is at-worst-nodal of relative dimension $1$ and such that formation of $S'$ commutes with arbitrary base change. By More on Morphisms of Spaces, Lemma 76.55.4 there is a maximal open subspace $X' \subset X$ such that $f|_{X'} : X' \to S$ is at-worst-nodal of relative dimension $1$. Moreover, formation of $X'$ commutes with base change. Hence we can take

\[ S' = S \setminus |f|(|X| \setminus |X'|) \]

This is open because a proper morphism is universally closed by definition. $\square$

The Stacks project

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