Lemma 33.26.2. Let $X$ be a proper variety over $k$. Then

1. $K = H^0(X, \mathcal{O}_ X)$ is a field which is a finite extension of the field $k$,

2. if $X$ is geometrically reduced, then $K/k$ is separable,

3. if $X$ is geometrically irreducible, then $K/k$ is purely inseparable,

4. if $X$ is geometrically integral, then $K = k$.

Proof. This is a special case of Lemma 33.9.3. $\square$

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