109.12 Gorenstein curves
There is an open substack of $\mathcal{C}\! \mathit{urves}$ parametrizing the Gorenstein “curves”.
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
the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{Gorenstein}$,
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
the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{Gorenstein}$,
$X$ 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.
$\square$
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
the classifying morphism $S \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{Gorenstein, 1}$,
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
the classifying morphism $\mathop{\mathrm{Spec}}(k) \to \mathcal{C}\! \mathit{urves}$ factors through $\mathcal{C}\! \mathit{urves}^{Gorenstein, 1}$,
$X$ 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.
$\square$
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