Lemma 53.23.3. Let $k$ be a field. Let $X$ be a proper scheme over $k$ having dimension $1$ and $H^0(X, \mathcal{O}_ X) = k$. Assume the singularities of $X$ are at-worst-nodal. Let $C \subset X$ be a rational bridge (Example 53.23.1). For any field extension $K/k$ the base change $C_ K \subset X_ K$ is a finite disjoint union of rational bridges.
Proof. Let $k''/k'/k$ be as in the example. Since $k'/k$ is finite separable, we see that $k' \otimes _ k K = K'_1 \times \ldots \times K'_ n$ is a finite product of finite separable extensions $K'_ i/K$. The corresponding product decomposition $k'' \otimes _ k K = \prod K''_ i$ gives degree $2$ separable algebra extensions $K''_ i/K'_ i$. Set $C_ i = C_{K'_ i}$. Then $C_ K = \coprod C_ i$ and therefore each $C_ i$ has genus $0$ (viewed as a curve over $K'_ i$), because $H^1(C_ K, \mathcal{O}_{C_ K}) = 0$ by flat base change. Finally, we have $X'_ K \cap C_ i = \mathop{\mathrm{Spec}}(K''_ i)$ has degree $2$ over $K'_ i$ as desired. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: