Lemma 53.22.3. 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 tail (Example 53.22.1). For any field extension $K/k$ the base change $C_ K \subset X_ K$ is a finite disjoint union of rational tails.
Proof. Let $x \in C$ and $k' = \kappa (x)$ be as in the example. Observe that $C \cong \mathbf{P}^1_{k'}$ by Proposition 53.10.4. 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 $K'_ i/K$. Set $C_ i = \mathbf{P}^1_{K'_ i}$ and denote $x_ i \in C_ i$ the inverse image of $x$. Then $C_ K = \coprod C_ i$ and $X'_ K \cap C_ i = x_ i$ as desired. $\square$
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