Chern Classes of $\mathcal{O}_E(1)$ on $\mathbb{P}(E)$ for $E = \mathcal{O} \oplus \mathcal{O}(n) \to \mathbb{P}^2$

Question

Let $E =\mathcal{O} \oplus \mathcal{O}(n) \to \mathbb{P}^2$ and denote by $\mathcal{O}_E(1)$ the dual of the tautological bundle.

How can I compute $c_1^2(\mathcal{O}_E(1)), c_1^3(\mathcal{O}_E(1))$, $c_1(\mathcal{O}_E(1))c_2(\mathcal{O}_E(1))$, and $c_3(\mathcal{O}_E(1))$?

I know that in the case of Hirzebruch surfaces, $c_1^2 = -n$, but I don't know how to tackle this problem.

Answer 1

Use the projective bundle formula in the Chow ring of $\mathbb{P}(E)$, see section "Cohomology ring and Chow group" of https://en.wikipedia.org/wiki/Projective_bundle.