If $X$ is a smooth projective curve over a field $k$, $V$ is a subbundle of some finite direct sum of $O\_X$, then is it true that $V$ is of degree 0?



degree 0 vector bundles
No. Counterexample: On $\mathbf P^1$, take a (two-element) basis of $\Gamma(\mathbf P^1, \mathcal O(1))$, so that the corresponding homomorphism $\mathcal O^2\to\mathcal O(1)$ is surjective. Its kernel is a subbundle of degree $-1$.


