The Stacks project

Lemma 53.23.6. Let $k$ be a field. Let $X$ be a proper scheme over $k$ of dimension $1$ with $H^0(X, \mathcal{O}_ X) = k$ having genus $g \geq 2$. Assume the singularities of $X$ are at-worst-nodal and that $X$ has no rational tails. Consider a sequence

\[ X = X_0 \to X_1 \to \ldots \to X_ n = X' \]

of contractions of rational bridges (Example 53.23.1) until none are left. Then $\omega _{X'}$ ample. The morphism $X \to X'$ is independent of choices and formation of this morphism commutes with base field extensions.

Proof. We proceed by contracting rational bridges until there are none left. Then $\omega _{X'}$ is ample by Lemma 53.23.5.

Denote $f : X \to X'$ the composition. By Lemma 53.23.4 and induction we see that $f^*\omega _{X'} = \omega _ X$. We have $f_*\mathcal{O}_ X = \mathcal{O}_{X'}$ because this is true for contraction of a rational bridge. Thus the projection formula says that $f_*f^*\mathcal{L} = \mathcal{L}$ for all invertible $\mathcal{O}_{X'}$-modules $\mathcal{L}$. Hence

\[ \Gamma (X', \omega _{X'}^{\otimes m}) = \Gamma (X, \omega _ X^{\otimes m}) \]

for all $m$. Since $X'$ is the Proj of the direct sum of these by Morphisms, Lemma 29.43.17 we conclude that the morphism $X \to X'$ is completely canonical.

Let $K/k$ be an extension of fields, then $\omega _{X_ K}$ is the pullback of $\omega _ X$ (Lemma 53.4.4) and we have $\Gamma (X, \omega _ X^{\otimes m}) \otimes _ k K$ is equal to $\Gamma (X_ K, \omega _{X_ K}^{\otimes m})$ by Cohomology of Schemes, Lemma 30.5.2. Thus formation of $f : X \to X'$ commutes with base change by $K/k$ by the arguments given above. Some details omitted. $\square$

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