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

1. 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.

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

3. 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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