Lemma 53.18.4. Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. If $X' \to X$ is a birational proper morphism, then
If $X$ is reduced, $H^0(X, \mathcal{O}_ X) \to H^0(X', \mathcal{O}_{X'})$ is surjective, and equality holds, then $X' = X$.
Proof. If $f : X' \to X$ is proper birational, then it is finite and an isomorphism over a dense open (see Varieties, Lemmas 33.17.2 and 33.17.3). Thus the inequality by Lemma 53.18.3. Assume $X$ is reduced. Then $\mathcal{O}_ X \to f_*\mathcal{O}_{X'}$ is injective and we obtain a short exact sequence
Under the assumptions given in the second statement, we conclude from the long exact cohomology sequence that $H^0(X, \mathcal{F}) = 0$. Then $\mathcal{F} = 0$ because $\mathcal{F}$ is generated by global sections (Varieties, Lemma 33.33.3). and $\mathcal{O}_ X = f_*\mathcal{O}_{X'}$. Since $f$ is affine this implies $X = X'$. $\square$
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)