Let $C$ be a smooth projective curve and $E$ a vector bundle of rank $r$ on $C$. We say that $E$ is nef/ample if $\mathcal{O}\_{\mathbb{P}(E)}(1)$ is so. Equivalently (see Hartshorne's papers on 'Ample vector bundles' and 'Ample vector bundles on curves'), $E$ is ample if and only if for any coherent $F$, $S^m(E)\otimes F$ is globally generated for all $m\geq n\_0$.


The statement I'm slightly stuck on is the following comment in a paper of Fujita 'On Kahler fibre spaces over curves': If $C$ has genus $g\geq2$ and $H^1(C, E)=0$ for some vector bundle $E$, then $E$ is ample.


This follows easily in the rank $1$ case from Riemann-Roch. I suspect the general case will also be easy but I have been through Lazarsfeld's book and the standard references with no luck so far. Any help appreciated!



Ample vector bundles on curves
Here's a partial answer. Suppose we're in characteristic 0 (Fujita would be assuming this),
and that $rank(E)=2$. By cor 7.6 of Hartshorne's ample vector bundles paper, it suffices
to check that $deg(E)>0$ and $deg(L)>0$ for ay quotient line bundle.
From Riemann-Roch as in Piotr's comment, we get
$$deg(E) + rank(E)(1-g) = h^0(E)\ge 0$$
which implies positivity of $deg(E)$. On a curve $H^1(E)=0$ implies the vanishing for
any quotient bundle, and so in particular for $L$. Combing this with the above argument, gives
$deg(L)>0$.


I think this can be pushed, but I'd better back to the less fun things that I'm supposed to be doing now.


