Lemma 33.42.7. 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 33.42.5. By Lemma 33.37.4 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 30.19.2). We obtain a corresponding morphism

such that the inverse image of $D_{+}(T_0)$ is $X$, see Constructions, Lemma 27.13.1. 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 30.21.2^{1} we conclude that $f$ is finite. Hence $X = f^{-1}(D_{+}(T_0))$ is affine.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: