Archimedean fibers "intersecting" curves on arithmetic surfaces

Question

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 $C$.

Answer 1

It seems that you are beginning to study Arakelov geometry. In this theory, the archimedean fibers have to be “thought of” fibers, but they can't be in the strict sense of scheme theory.

In fact, in Arakelov theory, the vertical fiber is never treated as a closed subscheme. Rather, the usual cycles on the arithmetic scheme are enhanced with an archimedean structure (Green currents) which is used to define their intersection as an enhanced cycle.

In Arakelov's theory for surfaces, these Green currents are canonically chosen; however, the general theory of Gillet-Soulé relaxes this assumption for it is useful to be able to pick Green currents according to one's choice (or to what geometry dictates).