Lemma 33.10.7. Let k be a field. Let X be a proper geometrically normal scheme over k. The following are equivalent
H^0(X, \mathcal{O}_ X) = k,
X is geometrically connected,
X is geometrically irreducible, and
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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