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

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 67.33.2),

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 $1$.