Lemma 53.13.1. Let $k$ be a field. Let $f : X \to Y$ be a surjective morphism of curves over $k$. If $X$ is smooth over $k$ and $Y$ is normal, then $Y$ is smooth over $k$.

**Proof.**
Let $y \in Y$. Pick $x \in X$ mapping to $y$. By Varieties, Lemma 33.25.9 it suffices to show that $f$ is flat at $x$. This follows from Lemma 53.2.3.
$\square$

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