This is a pretty basic question. Hartshorne defines "intersection multiplicity" for any two divisors on a surface. Fulton has an impressive framework of generalizing this in his book (my understanding of which is scant). But for whatever reason, in arithmetic texts one often sees intersection multiplicities defined only between a (general) divisor and a vertical divisor. What's really going on? Does this just make it easier to explain things, or is there an actual impediment to defining an intersection number between general divisors in the arithmetic setting? And if so, how does Fulton address it?
(just to be clear, when I say the "arithmetic setting" I mean in a scheme which has relative dimension 1 over a regular scheme of pure dimension 1)
Best Answer
When you intersect two divisor, you obtain a algebraic cycle of codimension 2. For a smooth surface, this is a collection of points that is well-defined up to rational equivalence. Now, if the surface is also proper ('compact'), you can count these points i.e. the number of points is the same for all representatives in the equivalence class.
The problem with arithmetical surfaces is that they are not compact! So you cannot apply the standard theory here. As far as I know, one usually tries to compactify arithmetic schemes using infinite points and Arakelov geometry. If one wants to avoid these matters, he has to put a restriction on divisors.