If $C$ and $D$ are smooth projective curves and $f:C\to D$ is a map, how do we compute $c_1(f_{*}\mathscr{O}_C)$?
I think applying Grothendieck Riemann-Roch from wikipedia yields the answer, but I'm hoping that is overkill, since I don't really understand GRR.
If $f$ is non-constant, it is finite and let $d=\deg f$. Then, $f_*\mathcal{O}_C=V$ is a rank $d$ vector bundle on $D$ and thus $\chi(V)=c_1(V)+d(1-g(D))$ by RR. On the other hand, $\chi(V)=\chi(\mathcal{O}_C)=1-g(C)$. So, $c_1(V)=(1-g(C))-d(1-g(D))$.