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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 $\dim X$)?

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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

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  • $\begingroup$ Great! Unfortunately, the application I'm looking at has r >= 3. $\endgroup$
    – Drew
    Jan 29, 2016 at 19:10
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    $\begingroup$ For future readers: the r=2 statement is Prop. 1.4 in the Ellia paper. $\endgroup$
    – Drew
    Jan 29, 2016 at 19:31

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