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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!

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Well, the cohomology/intersection theory of $\mathbb{CP}^2$ is rather simple. Its generated by the class of the hyperplane. So every curve is $dH$ where $H$ is the class of the hyperplane and $d$ is an integer.

Furthermore, $\mathbb P^2$ has no curves of class $dH$ for $d<0$. You've described a cubic: so its class is $3H$, and self intersection is $9H^2$.

So $H^2 = pt$. Either by your favourite argument from topology for the cup product or by observing that two (generic) lines in $\mathbb P^2$ intersect at a point transversally.

In general, actually finding the Poincare dual to some class and integrating can be a bit of a pain. $\mathbb P^2$ is actually not too bad though. I've essentially described it above in very vague terms. There's only one real class to think about which is the class of the hyperplane.

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One has $g=(d-1)(d-2)/2$ for smooth curves so your curve is necessarily singular, as you point out. But you don't need an immersion to calculate self intersection. Since the homology class of the curve is $3[\mathbb{C}P^1]$, its selfintersection is necessarily $9$. This is because a projective line has self-intersection $1$ by definition of projective geometry!

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