Lemma 32.42.4. Let $X$ be a proper scheme over a field $k$. If $\dim (X) \leq 1$ then $X$ is H-projective over $k$.

**Proof.**
By Lemma 32.42.3 we see that $X$ is a locally closed subscheme of $\mathbf{P}^ n_ k$ for some field $k$. Since $X$ is proper over $k$ it follows that $X$ is a closed subscheme of $\mathbf{P}^ n_ k$ (Morphisms, Lemma 28.39.7).
$\square$

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