The Stacks project

Lemma 33.43.9. Let $X$ be a curve over $k$. Then either $X$ is an affine scheme or $X$ is H-projective over $k$.

Proof. Choose $X \to \overline{X}$ with $\overline{X} \setminus X = \{ x_1, \ldots , x_ r\} $ as in Lemma 33.43.6. Then $\overline{X}$ is a curve as well. If $r = 0$, then $X = \overline{X}$ is H-projective over $k$. Thus we may assume $r \geq 1$ and our goal is to show that $X$ is affine. By Lemma 33.38.2 it suffices to show that $\overline{X} \setminus \{ x_1\} $ is affine. This reduces us to the claim stated in the next paragraph.

Let $X$ be an H-projective curve over $k$. Let $x \in X$ be a closed point such that $\mathcal{O}_{X, x}$ is a discrete valuation ring. Claim: $U = X \setminus \{ x\} $ is affine. By Lemma 33.43.8 the point $x$ defines an effective Cartier divisor of $X$. For $n \geq 1$ denote $nx = x + \ldots + x$ the $n$-fold sum, see Divisors, Definition 31.13.6. Denote $\mathcal{O}_{nx}$ the structure sheaf of $nx$ viewed as a coherent module on $X$. Since every invertible module on the local scheme $nx$ is trivial the first short exact sequence of Divisors, Remark 31.14.11 reads

\[ 0 \to \mathcal{O}_ X \xrightarrow {1} \mathcal{O}_ X(nx) \to \mathcal{O}_{nx} \to 0 \]

in our case. Note that $\dim _ k H^0(X, \mathcal{O}_{nx}) \geq n$. Namely, by Lemma 33.33.3 we have $H^0(X, \mathcal{O}_{nx}) = \mathcal{O}_{X, x}/(\pi ^ n)$ where $\pi $ in $\mathcal{O}_{X, x}$ is a uniformizer and the powers $\pi ^ i$ map to $k$-linearly independent elements in $\mathcal{O}_{X, x}/(\pi ^ n)$ for $i = 0, 1, \ldots , n - 1$. We have $\dim _ k H^1(X, \mathcal{O}_ X) < \infty $ by Cohomology of Schemes, Lemma 30.19.2. If $n > \dim _ k H^1(X, \mathcal{O}_ X)$ we conclude from the long exact cohomology sequence that there exists an $s \in \Gamma (X, \mathcal{O}_ X(nx))$ which is not a section of $\mathcal{O}_ X$. If we take $n$ minimal with this property, then $s$ will map to a generator of the stalk $\left(\mathcal{O}_ X(nx)\right)_ x$ since otherwise it would define a section of $\mathcal{O}_ X((n - 1)x) \subset \mathcal{O}_ X(nx)$. For this $n$ we conclude that $s_0 = 1$ and $s_1 = s$ generate the invertible module $\mathcal{L} = \mathcal{O}_ X(nx)$.

Consider the corresponding morphism $f = \varphi _{\mathcal{L}, (s_0, s_1)} : X \to \mathbf{P}^1_ k$ of Constructions, Section 27.13. Observe that the inverse image of $D_{+}(T_0)$ is $U = X \setminus \{ x\} $ as the section $s_0$ of $\mathcal{L}$ only vanishes at $x$. In particular, $f$ is non-constant, i.e., $\mathop{\mathrm{Im}}(f)$ has more than one point. Hence $f$ must map the generic point $\eta $ of $X$ to the generic point of $\mathbf{P}^1_ k$. Hence if $y \in \mathbf{P}^1_ k$ is a closed point, then $f^{-1}(\{ y\} )$ is a closed set of $X$ not containing $\eta $, hence finite. Finally, $f$ is proper1. By Cohomology of Schemes, Lemma 30.21.22 we conclude that $f$ is finite. Hence $U = f^{-1}(D_{+}(T_0))$ is affine. $\square$

[1] Namely, a H-projective variety is a proper variety by Morphisms, Lemma 29.43.13. A morphism of varieties whose source is a proper variety is a proper morphism by Morphisms, Lemma 29.41.7.
[2] One can avoid using this lemma which relies on the theorem of formal functions. Namely, $X$ 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 29.44.7.

Comments (2)

Comment #6631 by WhatJiaranEatsTonight on

The proof use 37.4 to find the globally generated invertible sheaf and the section s. But the assumption of 37.4 need X to be affine. And at last, you prove X is affine.

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  • 4 comment(s) on Section 33.43: Curves

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