Let $f:X\rightarrow Y$ be a finite morphism of non-singular projective varieties of degree $d$.
- Consider the map of sheaves $O_Y\rightarrow f_*O_X$. Is this morphism injective? Why? $f_*O_X$ is a rank $d$ vector bundle.
- Suppose the above morphism of sheaves is injective. Consider the exact sequence where $L$ is the cokernel: $$0\rightarrow O_Y\rightarrow f_*O_X\rightarrow L\rightarrow 0.$$ Does this exact sequence split and give us $ f_*O_X=O_Y\oplus L$?
- Suppose $f$ is of degree 2 and $X$ and $Y$ are non-singular surfaces. Let $C\subset Y$ be a non-singular curve along which $f$ is branched. Then is there is some relation between $L$ (as above ) and $O_Y(C)$?