The Stacks project

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

\[ \dim _ k H^1(X, \mathcal{O}_ X) \geq \dim _ k H^1(X', \mathcal{O}_{X'}) \]

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

\[ 0 \to \mathcal{O}_ X \to f_*\mathcal{O}_{X'} \to \mathcal{F} \to 0 \]

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 $X = X'$. $\square$

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