The Stacks project

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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0E6M. Beware of the difference between the letter 'O' and the digit '0'.