The Stacks project

Lemma 55.14.4. Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. Assume $X \to \mathop{\mathrm{Spec}}(R)$ is at-worst-nodal of relative dimension $1$ over $R$. Let $X \to X'$ be the contraction of an exceptional curve $E \subset X$ of the first kind. Then $X'$ is at-worst-nodal of relative dimension $1$ over $R$.

Proof. Namely, let $x' \in X'$ be the image of $E$. Then the only issue is to see that $X' \to \mathop{\mathrm{Spec}}(R)$ is at-worst-nodal of relative dimension $1$ in a neighbourhood of $x'$. The closed fibre of $X \to \mathop{\mathrm{Spec}}(R)$ is reduced, hence $\pi \in R$ vanishes to order $1$ on $E$. This immediately implies that $\pi $ viewed as an element of $\mathfrak m_{x'} \subset \mathcal{O}_{X', x'}$ but is not in $\mathfrak m_{x'}^2$. Since $\mathcal{O}_{X', x'}$ is regular of dimension $2$ (by definition of contractions in Resolution of Surfaces, Section 54.16), this implies that $\mathcal{O}_{X'_ k, x'}$ is regular of dimension $1$ (Algebra, Lemma 10.106.3). On the other hand, the curve $E$ has to meet at least one other component, say $C$ of the closed fibre $X_ k$. Say $x \in E \cap C$. Then $x$ is a node of the special fibre $X_ k$ and hence $\kappa (x)/k$ is finite separable, see Algebraic Curves, Lemma 53.19.7. Since $x \mapsto x'$ we conclude that $\kappa (x')/k$ is finite separable. By Algebra, Lemma 10.140.5 we conclude that $X'_ k \to \mathop{\mathrm{Spec}}(k)$ is smooth in an open neighbourhood of $x'$. Combined with flatness, this proves that $X' \to \mathop{\mathrm{Spec}}(R)$ is smooth in a neighbourhood of $x'$ (Morphisms, Lemma 29.34.14). This finishes the proof as a smooth morphism of relative dimension $1$ is at-worst-nodal of relative dimension $1$ (Algebraic Curves, Lemma 53.20.3). $\square$


Comments (0)

There are also:

  • 1 comment(s) on Section 55.14: Semistable reduction

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Your email address will not be published.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