# Degree of irreducible locally free sheaves and global sections on curves

Let $X$ be a smooth projective curve and $\mathcal{F}$ a locally free sheaf on $X$ of rank $2$ and negative degree. Assume further that $\mathcal{F}$ is irreducible in the sense, $\mathcal{F}$ cannot be written as a direct sum of two line bundles. Is it then true that $\mathcal{F}$ has no global sections?

Let $X$ be a smooth projective curve of genus 2 defined over an algebraically closed field $k$ of characteristic 0, and let $p \in X$ be given. Then $${\rm Ext}^{1}(\mathcal{O}_{X}(-p),\mathcal{O}_{X}) \cong H^{1}(\mathcal{O}_{X}(p)) \cong H^{0}(\omega_{X}(-p))^{\ast} \cong k$$ It follows that there is a nonsplit exact sequence $$0 \rightarrow \mathcal{O}_{X} \rightarrow E \rightarrow \mathcal{O}_{X}(-p) \rightarrow 0$$ where $E$ is a rank-2 vector bundle on $X.$ Clearly $h^{0}(E) =1$ and $c_{1}(E) = -1.$ I claim that $E$ cannot be written as a direct sum of line bundles. Indeed, if $E \cong L_{1} \oplus L_{2}$ where $c_{1}(L_1)=-c_{1}(L_2)-1,$ (wlog, $c_{1}(L_{1}) < c_{1}(L_{2})$) then we must have $c_{1}(L_{2}) \geq 0.$ If $c_{1}(L_{2}) > 0,$ then the cokernel of $\mathcal{O}_{X} \rightarrow E$ has torsion, which is impossible; thus $c_{1}(L_2)=0$, and $L_2 \cong \mathcal{O}_{X}.$ In addition, we have that $c_{1}(L_{1})=-1,$ and $E \rightarrow \mathcal{O}_{X}(-p)$ induces a surjection $L_{1} \rightarrow \mathcal{O}_{X}(-p),$ which must be an isomorphism. This contradicts the fact that our extension is nonsplit.