Lemma 33.9.5. Let $k$ be a field. Let $X$ be a proper geometrically reduced scheme over $k$. The following are equivalent

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

$X$ is geometrically connected.

Lemma 33.9.5. Let $k$ be a field. Let $X$ be a proper geometrically reduced scheme over $k$. The following are equivalent

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

$X$ is geometrically connected.

**Proof.**
By Lemma 33.9.4 we have (1) $\Rightarrow $ (2). By Lemma 33.9.3 we have (2) $\Rightarrow $ (1).
$\square$

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