Assume we are over $\mathbb C$. Let $C$ be a complete algebraic curve, and $E$ an algebraic vector bundle. Its Hilbert polynomial is
$$p(t)=rt+r(1-g)+d$$
where $r=\mathrm{rank}(E)$ and $d=\deg(E)$ and $g$ is the genus of $C$. So a condition on the Hilbert polynomial is a condition on the rank, degree and genus.


It is known that, if $r=1$ and $d<0$, then $E$ has no nonzero section. I wonder if there are similar conditions for general rank.



For a vector bundle over a curve, is there a condition on the Hilbert polynomial for no non-zero section?
If the bundle is semi-stable, then $H^0(E) \neq 0$ implies $\deg(E) \geq 0$. Indeed, a global section of $E$ induces a non-zero map of shaves $\mathcal{O}\_C \to E$, and semi-stability implies $$0= \mu (\mathcal{O}\_C) \leq \mu(E)=\frac{1}{r}\deg(E).$$ For strictly unstable bundles one cannot say anything. For instance, if $L$ is a line bundle of degree $-1$ on $C$, then $E=L^{\otimes n} \oplus \mathcal{O}\_C$ has degree $-n$ and $H^0(E)=1$, so for all $g \geq 0$ and for all $r \geq 2$ there exist unstable vector bundles on $C$ with non-zero global sections and arbitrarily negative degree.


