The Stacks project

Definition 53.19.1. Let $k$ be a field. Let $X$ be a $1$-dimensional locally algebraic $k$-scheme.

  1. We say a closed point $x \in X$ is a node, or an ordinary double point, or defines a nodal singularity if there exists an ordinary double point $\overline{x} \in X_{\overline{k}}$ mapping to $x$.

  2. We say the singularities of $X$ are at-worst-nodal if all closed points of $X$ are either in the smooth locus of the structure morphism $X \to \mathop{\mathrm{Spec}}(k)$ or are ordinary double points.

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View Definition 53.19.1 as pdf