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Dec 3, 2014 at 12:46 comment added diverietti Hi Angelo, maybe there is something that I am missing, but I cannot understand completely your last argument. Why do you mention the fact that every extension of line bundles on $\mathbb P^2$ splits? In any case, if $E$ fits in a short exact sequence of vector bundles, say with left and right terms respectively $\mathcal O(a)$ and $\mathcal O(b)$, then its Chern classes are given respectively by $a+b$ and $ab$, and one may certainly choose $k$ and $m$ to be not of that form, isn't it?
Jan 22, 2012 at 20:11 vote accept Botong Wang
Jan 18, 2012 at 5:02 history edited Angelo CC BY-SA 3.0
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Jan 17, 2012 at 19:45 history edited Angelo CC BY-SA 3.0
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Jan 17, 2012 at 19:33 history answered Angelo CC BY-SA 3.0