Let $X$ be a smooth projective variety of dimension $n$, and let $\pi: C \to \mathbb{P}^1$ be a cover of the projective line, ramified at some points $S=\{p_1, \ldots, p_r\}$. Consider the morphism
$f: X \times C \to \mathbb{P}^1$
obtained by composition of the projection to $C$ with $\pi$.
I am wondering what can be said about the local systems $R^m f_\ast \mathbb{Q}$ on $U=\mathbb{P}^1-S.$
Are they interesting?
Can one compute their monodromy in terms of the monodromy of $\pi$?
What are the cohomologies $H^k_c(U, R^m f_\ast \mathbb{Q})$ for $k=0$ and $1$?
Any help will be appreciated, thanks!