I am reading Petitjean's thesis (1995). I have trouble making sense of the following:
Let $\mathcal{E}$ be a vector bundle (locally free sheaf) of rank $2$ over a curve $C$ with genus $g$ and let $Z = P(\mathcal{E})$. The ruled surface $R$ is defined by the image $R=\rho(Z)$ by some map $\rho:Z\rightarrow\mathbb{P}^3$ that is birational and maps the fibers of $Z$ onto lines in $\mathbb{P}^3$.
Let $\mu_0$ be the degree of $R$, $\pi:Z\rightarrow C$ the projective morphism (bundle map) of $Z$ and for $x\in C$ let $F = \pi^{-1}(x)$. Let $f$ be the class of $F$ in $A^1(Z)$. Then $f^2 = [\pi^{-1}(x)]^2 = \pi^*[x]^2 = 0$ and $$ \pi^*c_1(\mathcal{E}) = -\mu_0 f\\ \pi^*c_1(T_C) = (2-2g) f.$$
I can't make sense of these equations. $(2-2g)$ is the Euler characteristic and $c_1(T_C)$ is the Euler class of $C$, so I see that there is a connection. But I don't know how to compute the action of $\pi^*$.
Thank you for any help!