Archimedean fibers “intersecting” curves on arithmetic surfaces

Let's fix a number field $K$ with its ring of integers $O_K$. Moreover consider an arithmetic surface $f:S\to \text{Spec } O_K$. For every archimedean place $\sigma$ in $K$, $K_\sigma$ is the completion of $K$ with respect to $\sigma$ (note that $K_\sigma$ is $\mathbb R$ or $\mathbb C$).

We can imagine $\sigma$ as "a point at infinity" of $\text{Spec } O_K$, and therefore we define the archimedean fiber over $\sigma$ as: $$S_\sigma:= S\times_{O_K}\text{Spec }K_\sigma$$

I'd like to understand how $S_\sigma$ interacts with $S$ and the (integral) curves on $S$:

1. Is the canonical morphism $S_\sigma\to S$ a closed embedding? I mean: what can we really recover of $S_\sigma$ on $S$? I would be surprised if $S_\sigma$ could be embedded in $S$.
2. Let $C\subset S$ be an integral vertical curve on $S$, then my intuition says that "the intersection $S_\sigma\cap C$" should be empty. Is it true?
3. On the contrary if $C$ is horizontal, "the intersection $S_\sigma\cap C$" should be non empty. Is it true?

Edit: Here by $S_\sigma\cap C$ I mean the set of points of $S_\sigma$ mapped in $C$. In other words, they are the points at infinity of $C$.

• Your definition of archimedean fiber doesn't seem to match with the convention for non-archimedean places. In particular, when $\sigma$ is non-archimedean, the fiber $S_\sigma$ you define is generic rather than special. A true archimedean fiber needs to live outside traditional scheme theory. – S. Carnahan May 8 '16 at 16:05