global sections of symmetric powers of vector bundle of rank 2

Question

Consider on a smooth projective curve $X$,we have a degree nonnegative vector bundle $E$ of rank two, can we imply there exists some $m\in\mathbb N$, such that $H^0(X,\textit{Sym}^m(E))\not= 0$?

For the case of $\textit{genus}(C)=0$, this can be easily deduced from Riemann--Roch formula and take m=1. But can we say prove the case for higher genus?

Answer 1

This is not true --- take $E = L_1 \oplus L_2$, where $L_1$ and $L_2$ are line bundles of degree zero such that the corresponding points of $\mathrm{Pic}^0(X)$ are linearly independent over $\mathbb{Q}$. Then for each $m$ the bundle $\mathrm{Sym}^m(E)$ is a direct sum of nontrivial degree zero line bundles, hence it has no global sections.