Definition 33.10.1. Let $X$ be a scheme over the field $k$.

Let $x \in X$. We say $X$ is

*geometrically normal at $x$*if for every field extension $k \subset k'$ and every $x' \in X_{k'}$ lying over $x$ the local ring $\mathcal{O}_{X_{k'}, x'}$ is normal.We say $X$ is

*geometrically normal*over $k$ if $X$ is geometrically normal at every $x \in X$.

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