Lemma 53.19.11. Let $k$ be a field. Let $X$ be a $1$-dimensional algebraic $k$-scheme. Let $x \in X$ be a closed point. The following are equivalent
$x$ is a split node,
$x$ is a node and there are exactly two points $x_1, x_2$ of the normalization $X^\nu $ lying over $x$ with $k = \kappa (x_1) = \kappa (x_2)$,
$\mathcal{O}_{X, x}^\wedge \cong k[[x, y]]/(xy)$ as a $k$-algebra, and
add more here.
Proof. This follows from the discussion in Remark 53.19.9 and Lemma 53.19.7. $\square$
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