Lemma 109.12.2 (0E6M)—The Stacks project

Lemma 109.12.2. There exist an open substack $\mathcal{C}\! \mathit{urves}^{Gorenstein, 1} \subset \mathcal{C}\! \mathit{urves}$ 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 $\mathcal{C}\! \mathit{urves}^{Gorenstein, 1}$,
2. the morphism $X \to S$ is Gorenstein and has relative dimension $1$ (Morphisms of Spaces, Definition 67.33.2),
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 $\mathcal{C}\! \mathit{urves}^{Gorenstein, 1}$,
2. $X$ is Gorenstein and $X$ is equidimensional of dimension $1$.

Proof. Recall that a Gorenstein scheme is Cohen-Macaulay (Duality for Schemes, Lemma 48.24.2) and that a Gorenstein morphism is a Cohen-Macaulay morphism (Duality for Schemes, Lemma 48.25.4. Thus we can set $\mathcal{C}\! \mathit{urves}^{Gorenstein, 1}$ equal to the intersection of $\mathcal{C}\! \mathit{urves}^{Gorenstein}$ and $\mathcal{C}\! \mathit{urves}^{CM, 1}$ inside of $\mathcal{C}\! \mathit{urves}$ and use Lemmas 109.12.1 and 109.8.2. $\square$

The Stacks project

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