Suppose you have a surface (of general type if it helps), and it's embedded in a 3-fold. Naively, the intersection of divisors on the 3 fold and the surface, will be a divisor (a curve) in a surface. I want to know whether this curve is reducible or not, and if it is, then how should I write it in terms of the reducible ones on the surface.
Does anyone can help me to find some reference for it? Usually the textbooks assume that we have reduced irreducible curve, but they don't say how to check that in general. Is there any criterion for that in terms of intersection numbers etc?
Thanks very much. Mohsen