Results 1 to 2 of 2

Math Help - Self-intersection of the pullback of a hyperplane

  1. #1
    Newbie
    Joined
    Jul 2008
    Posts
    18

    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?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Jul 2008
    Posts
    18

    Nothing?

    Really? No one has any ideas on this?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. pullback is not the product
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: November 23rd 2011, 04:08 PM
  2. Hyperplane
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: March 28th 2011, 06:04 AM
  3. Hyperplane
    Posted in the Calculus Forum
    Replies: 0
    Last Post: October 3rd 2010, 10:21 AM
  4. Hyperplane (Please help me!)
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: June 27th 2010, 07:42 AM
  5. property of pullback
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: June 10th 2010, 12:09 PM

Search Tags


/mathhelpforum @mathhelpforum