# Connection between complex curves and homology classes

1. Jul 25, 2008

### cduston

Hey all,
This question stems from Scorpan, "The Wild World of 4-Manifolds", pg 302 (and all through that chapter). He states that a random homogeneous polynomial of degree d in CP^2 with coordinates $$[z_0:z_1:z_2]$$ defines a complex curve C, with homology class $$[C]=d[CP^1]$$.

So I understand that the homology classes of curves in CP^2 would be connected to the classes of CP^1, but I am a little surprised by the extra factor which is the degree of the curve. Is that simply because if you have some degree 2 homogeneous polynomial (say $$p(z_0,z_1,z_2)=z_0^2+z_1^2+z_2^2$$ ) then the homology class of CP^1 would have to "loop around twice" to cover the entire curve?

Any response would be appreciated.