The Stacks project

Definition 28.15.4. Let $X$ be a scheme. Let $x \in X$. The number of branches of $X$ at $x$ is the number of branches of the local ring $\mathcal{O}_{X, x}$ as defined in More on Algebra, Definition 15.106.6. The number of geometric branches of $X$ at $x$ is the number of geometric branches of the local ring $\mathcal{O}_{X, x}$ as defined in More on Algebra, Definition 15.106.6.