Suppose we have a subvariety (codimension one) inside a smooth variety with a fixed divisor class D. However, let it be a singular one. I'm just wondering how much the usual techniques (such as adjunction, Riemann Roch etc.) for computing the cohomology and Chern classes are reliable in this situation, and what are the alternatives in such cases?
The second question is, suppose now I resolve the singularities, I know they are rational double points, then how should I compare the topology and cohomology of this resolved new space, with the smooth elements of the linear system of D?
Thanks very much,
Mohsen