Let $\pi: X\to C$ be a fibration in curves where $C$ is a non-singular curve and $X$ a regular, integral surface and the generic fiber $X_\eta$ is a non-singular curve over $k(C)$ (these hypotheses might be stronger than necessary, but I just threw a bunch on to make it as nice as possible).
Now a point on $X_\eta$, say $p$ is also a point on $X$ itself. Note the generic point of $X$ is the generic point of $X_\eta$, say $\zeta$. All the other points on $X_\eta$ are closed in the curve and are height $1$ points on $X$.
I read in a paper that for any point on the generic fiber, $p$, we have $\mathcal{O}_{X,p}\simeq \mathcal{O}_{X_\eta, p}$, and at first I just thought to myself that it's obvious, but when I tried to actually think of a reason it wasn't so obvious.
If $X_\eta$ were open in $X$, then this would be clear since restricting to an open and then taking a stalk doesn't cause problems, but why should this still be true for the generic fiber which is neither open nor closed?
One noted consequence is that $k(X)=\mathcal{O}_{X,\zeta}=\mathcal{O}_{X_\eta, \zeta}=k(X_\eta)$.