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 29.42.17 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:

  • 2 comment(s) on Section 53.23: Contracting rational bridges

Post a comment

Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0E3Q. Beware of the difference between the letter 'O' and the digit '0'.