I'm currently trying to get a grip on actually calculating some differential-geometric definitions. I'm looking at the following map from $\mathbb{CP}^{1}$ to $\mathbb{CP}^2$ :
$f([z_0,z_1])=[z_0^3,z_0 z_1^2,z_1^3]$
What I don't understand is how one would go about calculating the self-intersection number of this. It's not an immersion, so do I need to take an immersion in the same homology class first? And how do I actually calculate it - by perturbing the map a little and then counting transversal intersections with sign (this turned out to be very messy with my choices), or by actually finding the Poincaré dual (how would one go about that?) and integrating it?
In case anyone is wondering, this problem arose while trying to understand the adjunction inequality for J-holomorphic curves.
Thanks already for any help you can give!