Definition 53.16.2. Let $k$ be an algebraically closed field. Let $X$ be an algebraic $1$-dimensional $k$-scheme. Let $x \in X$ be a closed point. We say $x$ defines a *multicross singularity* if the completion $\mathcal{O}_{X, x}^\wedge $ is isomorphic to (53.16.0.1) for some $n \geq 2$. We say $x$ is a *node*, or an *ordinary double point*, or *defines a nodal singularity* if $n = 2$.

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