Let $p:C\to\mathbb{P}^1$ be a degree $k$ morphism from a smooth projective curve $C$ to the projective line and $L$ a very ample line bundle on $C$. We know that $p\_\*\mathcal{O}\_C(L)$ is a rank $k$ locally free sheaf on $\mathbb{P}^1$ and hence is in the form $\mathcal{O}(e\_1)\oplus\cdots\oplus\mathcal{O}(e\_k)$ by Birkhoff–Grothendieck theorem. In this case, are all $e\_1,...,e\_k$ positive?



Pushforward of a very ample line bundle on a curve to $\mathbb{P}^1$
No, not in general. Take $C=\mathbb{P}^1$, $L=\mathcal{O}(1)$, $p$ to be map $x\mapsto x^2$ in affine coordinates. Then $p\_\*L$ has rank $2$, but  

$$2=h^0(L)=h^0(p\_\*L)=h^0(\mathcal{O}(e\_1))+h^0(\mathcal{O}(e\_2))$$
If $e\_1$ and $e\_2$ were both positive, then term on the right would be at least $4$. So this is impossible.


**Added in response to comment.** If you are allowed to pick $\deg L\gg 0$ relative to $k$, then I think it's probably true. Here's a result in that direction.



> 
> Lemma. If $\deg L\gg 0$ relative to $k$, then all $e\_i\ge 0$.
> 
> 
> 


Sketch. We can assume $L=\omega\_{C/\mathbb{P}^1}(M)$ with $M$ globally generated. By a standard trick, we can find a cyclic cover $\pi:\tilde C\to C$ such that $L$ is a direct summand of $\pi\_\*\omega\_{\tilde C/\mathbb{P}^1}$. Then $p\_\*L$ is a summand of $(p\circ \pi)\_\*\omega\_{\tilde C/\mathbb{P}^1}$. The last sheaf is semipositive by a theorem of Fujita.


I suspect with more work, you can make the $e\_i$ positive, but I leave that to you.


