Lemma 109.12.1 (0E1M)—The Stacks project

Table of contents
Part 8: Topics in Moduli Theory
Chapter 109: Moduli of Curves
Section 109.12: Gorenstein curves
Lemma 109.12.1 (cite)

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Lemma 109.12.1. There exist an open substack $\mathcal{C}\! \mathit{urves}^{Gorenstein} \subset \mathcal{C}\! \mathit{urves}$ such that

given a family of curves $X \to S$ the following are equivalent
1. the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{Gorenstein}$,
2. the morphism $X \to S$ is Gorenstein,
given a scheme $X$ 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 $\mathcal{C}\! \mathit{urves}^{Gorenstein}$,
2. $X$ is Gorenstein.

Proof. Let $f : X \to S$ be a family of curves. By More on Morphisms of Spaces, Lemma 76.27.7 the set

\[ W = \{ x \in |X| : f \text{ is Gorenstein at }x\} \]

is open in $|X|$ and formation of this open commutes with arbitrary base change. Since $f$ is proper the subset

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

of $S$ is open and $X \times _ S S' \to S'$ is Gorenstein. Moreover, formation of $S'$ commutes with arbitrary base change because this is true for $W$ Thus we get the open substack with the desired properties by the method discussed in Section 109.6. $\square$

The Stacks project

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