# Self-intersection of the pullback of a hyperplane

• Oct 29th 2008, 12:58 PM
cduston
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 $Y_r$ of the complex projective plane $\mathbb{C}P^2$, with covering map $f$. I want to pull the hyperplane $[H]\subset\mathbb{C}P^2$ 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

$f^*([H])\cdot f^*([H])=d.$

In case it matters, I think that $[H]=[\mathbb{C}P^1]$ 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 $[\mathbb{C}P^1]\cdot[\mathbb{C}P^1]=1$. So in that case (since the pullback of the hyperplane over a covering of degree d should be $d[H]$), when I do this calculation I get

$f^*([H])\cdot f^*([H])=d^2 [\mathbb{C}P^1]\cdot[\mathbb{C}P^1]=d^2$

Can someone provide any thoughts, or possibly redirection if I am confused?
• Nov 3rd 2008, 08:28 AM
cduston
Nothing?
Really? No one has any ideas on this?