Lemma 32.42.3. Let $X$ be a separated finite type scheme over a field $k$. If $\dim (X) \leq 1$ then $X$ is H-quasi-projective over $k$.

**Proof.**
By Proposition 32.37.12 the scheme $X$ has an ample invertible sheaf $\mathcal{L}$. By Morphisms, Lemma 28.37.3 we see that $X$ is isomorphic to a locally closed subscheme of $\mathbf{P}^ n_ k$ over $\mathop{\mathrm{Spec}}(k)$. This is the definition of being H-quasi-projective over $k$, see Morphisms, Definition 28.38.1.
$\square$

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