TITLE: When is a general sheaf (on the projective plane) globally generated?
QUESTION [7 upvotes]: Let $v$ be a chern character on $\mathbb P^2$ so that the moduli of sheaves of chern character $v$ is non-empty of the expected dimension. When is it true that the general sheaf in moduli is globally generated?
Here is a first guess: One expects $s$ sections of a rank $r$ bundle to have rank at most $r-1$ in codimension $s-r+1$ (see e.g. the Wikipedia article on the Porteous formula). Hence, if $s \ge r+2$, we would expect the $s$ sections to generate. So:
Conjecture: If $\chi(\mathbb P^2, v) \ge r+2$ and the slope of $v$ is positive, then the general vector bundle with chern character $v$ is globally generated.
QUESTIONS
Is this true?
If not, what are some counterexamples? Is there a different theorem with this conclusion? 
If yes, does it work for other surfaces? How about for higher dimensional varieties (replacing $2$ with $\dim X$)?

REPLY [4 votes]: The result is true if  r=2 , see
Le Potier, J. Stabilité et amplitude sur P2. in Progress in Math., 7 (1980), 145–182, Birkhauser.
see also
Anghel, C., Coanda, I., Manolache, N. 
Globally Generated Vector Bundles on P^n with c_1=4
Ellia, P. 
Chern classes of rank two globally generated vector bundles on P2.
 Rend. Lincei Mat. Appl. 24 (2013), 147–163