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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