Self-intersection of the pullback of a hyperplane
Hey everyone,
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 ![[H]\subset\mathbb{C}P^2](http://latex.codecogs.com/png.latex?[H]\subset\mathbb{C}P^2)
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
![f^*([H])\cdot f^*([H])=d.](http://latex.codecogs.com/png.latex?f^*([H])\cdot f^*([H])=d.)
In case it matters, I think that ![[H]=[\mathbb{C}P^1]](http://latex.codecogs.com/png.latex?[H]=[\mathbb{C}P^1])
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 ![[\mathbb{C}P^1]\cdot[\mathbb{C}P^1]=1](http://latex.codecogs.com/png.latex?[\mathbb{C}P^1]\cdot[\mathbb{C}P^1]=1)
. So in that case (since the pullback of the hyperplane over a covering of degree d should be ![d[H]](http://latex.codecogs.com/png.latex?d[H])
), when I do this calculation I get
![f^*([H])\cdot f^*([H])=d^2 [\mathbb{C}P^1]\cdot[\mathbb{C}P^1]=d^2](http://latex.codecogs.com/png.latex?f^*([H])\cdot f^*([H])=d^2 [\mathbb{C}P^1]\cdot[\mathbb{C}P^1]=d^2)
Can someone provide any thoughts, or possibly redirection if I am confused?
