$f: X\longrightarrow Y$ is a totally ramified covering of degree d. It has ramified divisor $R \subset X$ and branch divisor $B \subset Y$. Suppose both X and Y are smooth. Then $f_*\mathcal{O}_X$ is locally free of rank d. But what is this locally free sheaf look like? Does it split into direct sum of line bundles ? How it relates the branch divisor in Y? As an example, we may think about plane curve $F_3 $defined by $z^3=x^3+y^3$ projects to $\mathbb{P}^1$, $f:F_3 \longrightarrow\mathbb{P}^1$, where f(x:y:z)=(x:y). Then f is of degree 3. Ramified divisor is $\{(-1:\omega_3:0),(-1:\omega_3^2:0),(-1:1:0)\}$, Branch divisor is $\{(-1:\omega_3),(-1:\omega_3^2),(-1:1)\}$. In this case what is $f_*\mathcal{O}_{F_3}$?
Thanks