Let $\mathcal E$ be a rank $n$ vector bundle over a curve $Y$ and let $X=\mathbb P(\mathcal E)$ and let $\pi: X \to Y$ be the projection. I would like to compute the value of the top self-intersection of the tautological line bundle $(\mathcal O\_X(1))^n$.



Top self-intersection of the tautological line bundle
Let $\xi \in A^1(X) $ be the class of $\mathcal{O}\_X(1)$.


Since $\dim Y=1$, by [Hartshorne, Algebraic Geometry, p. 429] we have the equality $$\xi^n=\pi^\*c\_1(\mathcal{E}) \cdot \xi^{n-1}.$$


But $\mathcal{O}\_X(1)$ is a relative hyperplane, so $\pi^\*(\textrm{point}) \cdot \xi^{n-1}=1$. Then $$\xi^n= \deg (\mathcal{E}).$$


