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 $\dim X$)?