Suppose $\pi:X\to C$ is a geometrically ruled surface, and $D$ a divisor on $X$. Then if $D.f=0$ for a fibre $f$, we know by Grauert's theorem that $\pi_{*}(\mathscr{L}(D))$ is a invertible sheaf on $C$.
It is know that $\pi^{*}\pi_{*}(\mathscr{L}(D))\simeq \mathscr{L}(D)$, can anyone help to explain why this is true?
Thanks!
