Let $R$ be a complete DVR with fraction field $K$, uniformizer $\pi$ and alg. closed residue field $k$.
Let $f : X\rightarrow \text{Spec }R$ be a prestable model of $\mathbb{P}^1_K$ with a $R$-section $\sigma\subset X$ lying in the smooth locus. Say $\sigma = \{\eta,\epsilon\}$, with $\eta\in X_K,\epsilon\in X_k$. Let $t$ be a coordinate on $\mathbb{P}^1_K$ such that $\eta$ is the point $t = 0$. Must there exist an open affine subscheme $U = \text{Spec }B$ of $X$ which contains $\sigma$ such that $\sigma$ is the divisor defined by $t$? In other words, is it true that:
(1) $B$, viewed as a subring of the function field $K(t)$, contains $t$, and
(2) $t$ generates, inside $B$, the ideal defining $\sigma$.
Intuitively, $t$ is a meromorphic function on $X$ which vanishes with order 1 at $\eta$, and so it should also vanish along $\sigma$ (the closure of $\eta$), which is to say it should exist in a suitable $B$, so I'm pretty confident (1) is true. I suspect (2) is false however, for the reason that presumably $t$ might vanish along the component of $X_k$ containing $\epsilon$.
