Definition 33.9.1. Let $X$ be a scheme over the field $k$.

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

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

*geometrically pointwise integral*if $X$ is geometrically pointwise integral at every point.We say $X$ is

*geometrically integral*over $k$ if the scheme $X_{k'}$ is integral for every field extension $k'$ of $k$.

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