Lemma 53.13.5. Let $k$ be a field of characteristic $p > 0$. Let $f : X \to Y$ be a nonconstant morphism of proper nonsingular curves over $k$. Assume

$X$ is smooth,

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

$k(X)/k(Y)$ is purely inseparable.

Then $Y$ is smooth, $H^0(Y, \mathcal{O}_ Y) = k$, and the genus of $Y$ is equal to the genus of $X$.

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