Lemma 33.10.7. Let $k$ be a field. Let $X$ be a proper geometrically normal scheme over $k$. The following are equivalent

1. $H^0(X, \mathcal{O}_ X) = k$,

2. $X$ is geometrically connected,

3. $X$ is geometrically irreducible, and

4. $X$ is geometrically integral.

Proof. By Lemma 33.9.5 we have the equivalence of (1) and (2). A locally Noetherian normal scheme (such as $X_{\overline{k}}$) is a disjoint union of its irreducible components (Properties, Lemma 28.7.6). Thus we see that (2) and (3) are equivalent. Since $X_{\overline{k}}$ is assumed reduced, we see that (3) and (4) are equivalent too. $\square$

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