## 107.12 Gorenstein curves

There is an open substack of $\mathcal{C}\! \mathit{urves}$ parametrizing the Gorenstein “curves”.

Lemma 107.12.1. There exist an open substack $\mathcal{C}\! \mathit{urves}^{Gorenstein} \subset \mathcal{C}\! \mathit{urves}$ such that

1. 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,

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 $\mathcal{C}\! \mathit{urves}^{Gorenstein}$,

2. $X$ is Gorenstein.

Proof. Let $f : X \to S$ be a family of curves. By More on Morphisms of Spaces, Lemma 74.27.7 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 107.6. $\square$

Lemma 107.12.2. There exist an open substack $\mathcal{C}\! \mathit{urves}^{Gorenstein, 1} \subset \mathcal{C}\! \mathit{urves}$ such that

1. 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 65.33.2),

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 107.12.1 and 107.8.2. $\square$

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