The Stacks project

Lemma 32.42.7. Let $X$ be a curve over $k$. Then either $X$ is an affine scheme or $X$ is H-projective over $k$.

Proof. Choose $X \to \overline{X}$ as in Lemma 32.42.5. By Lemma 32.37.4 we can find a globally generated invertible sheaf $\mathcal{L}$ on $\overline{X}$ and a section $s \in \Gamma (\overline{X}, \mathcal{L})$ such that $X = \overline{X}_ s$. Choose a basis $s = s_0, s_1, \ldots , s_ m$ of the finite dimensional $k$-vector space $\Gamma (\overline{X}, \mathcal{L})$ (Cohomology of Schemes, Lemma 29.19.2). We obtain a corresponding morphism

\[ f : \overline{X} \longrightarrow \mathbf{P}^ m_ k \]

such that the inverse image of $D_{+}(T_0)$ is $X$, see Constructions, Lemma 26.13.1. In particular, $f$ is non-constant, i.e., $\mathop{\mathrm{Im}}(f)$ has more than one point. A topological argument shows that $f$ maps the generic point $\eta $ of $\overline{X}$ to a nonclosed point of $\mathbf{P}^ n_ k$. Hence if $y \in \mathbf{P}^ n_ k$ is a closed point, then $f^{-1}(\{ y\} )$ is a closed set of $\overline{X}$ not containing $\eta $, hence finite. By Cohomology of Schemes, Lemma 29.21.21 we conclude that $f$ is finite. Hence $X = f^{-1}(D_{+}(T_0))$ is affine. $\square$

[1] One can avoid using this lemma which relies on the theorem of formal functions. Namely, $\overline{X}$ is projective hence it suffices to show a proper morphism $f : X \to Y$ with finite fibres between quasi-projective schemes over $k$ is finite. To do this, one chooses an affine open of $X$ containing the fibre of $f$ over a point $y$ using that any finite set of points of a quasi-projective scheme over $k$ is contained in an affine. Shrinking $Y$ to a small affine neighbourhood of $y$ one reduces to the case of a proper morphism between affines. Such a morphism is finite by Morphisms, Lemma 28.42.7.

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