Self-intersection of the pullback of a hyperplane
I am working on trying to figure out what is the self-intersection of the pullback of a hyperplane. In fact, I think I know the answer, I just don't know why! So here is the setup:
We have a degree d finite covering of the complex projective plane , with covering map . I want to pull the hyperplane back to the covering and find it's selfintersection. I know the answer is d, the degree of the covering map. So, in effect, I want to prove
In case it matters, I think that since the top homology group is trivial and the divisors should be equal up to linear equivalence. Although, that makes me even more confused because I was under the impression that . So in that case (since the pullback of the hyperplane over a covering of degree d should be ), when I do this calculation I get
Can someone provide any thoughts, or possibly redirection if I am confused?