The Stacks project

Lemma 109.23.5. Let $g \geq 2$. There is a morphism of algebraic stacks over $\mathbf{Z}$

\[ stabilization : \mathcal{C}\! \mathit{urves}^{prestable}_ g \longrightarrow \overline{\mathcal{M}}_ g \]

which sends a prestable family of curves $X \to S$ of genus $g$ to the stable family $Y \to S$ associated to it in Lemma 109.23.4.

Proof. To see this is true, it suffices to check that the construction of Lemma 109.23.4 is compatible with base change (and isomorphisms but that's immediate), see the (abuse of) language for algebraic stacks introduced in Properties of Stacks, Section 100.2. To see this it suffices to check properties (1) and (2) of Lemma 109.23.4 are stable under base change. This is immediately clear for (1). For (2) this follows either from the fact that the contractions of Algebraic Curves, Lemmas 53.22.6 and 53.23.6 are stable under ground field extensions, or because the conditions characterizing the morphisms on fibres in Algebraic Curves, Lemma 53.24.2 are preserved under ground field extensions. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 109.23: Contraction morphisms

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Your email address will not be published.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 0E8B. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 109.23.5 as pdf