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
Comments (0)
There are also: