Assume I have an onto morphism $\pi: Y\longrightarrow X$, where $X$ and $Y$ are both projective curves over an algebraically closed field $k$.
Also, assume that the following exact sequence of structure sheaves holds: $$0\longrightarrow\mathcal{O}_X\longrightarrow\pi_*\mathcal{O}_Y\longrightarrow k\longrightarrow 0$$
I would like to know whether the former exact sequence also implies that: $$0\longrightarrow\mathcal{O}_X(d)\longrightarrow\pi_*\mathcal{O}_Y(d)\longrightarrow k\longrightarrow 0$$
Holds for all $d$.