Lemma 53.2.5. Let $k$ be a field. Let $X \to Y$ be a morphism of varieties with $Y$ proper and $X$ a curve. There exists a factorization $X \to \overline{X} \to Y$ where $X \to \overline{X}$ is an open immersion and $\overline{X}$ is a projective curve.

Proof. This is clear from Lemma 53.2.1 and Varieties, Lemma 33.43.6. $\square$

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