# Do negative indecomposable bundles on curves have sections?

Let $$X$$ be a smooth projective curve, and $$E$$ an indecomposable vector bundle on $$X$$ with $$\mathrm{deg} E<0$$. Is it true that $$H^0(X,E)=0$$?

This is true if $$E$$ is a line bundle, which means it is also true whenever $$X$$ is $$\mathbb{P}^1$$, since all vector bundles split here.

It is also true by results of Atiyah if $$X$$ is an elliptic curve. What about for curves of higher genus?

The assumption that $$E$$ is indecomposable is of course necessary.