Given a non-semistable vector bundle on $P^2$, is there a way to tell what the rank P^2$ of rank 4, are explicit conditions known for when the maximal destabilizing bundle will beranks in its Harder-Narasimhan filtration are (3,1), (2,2) and (1,3) respectively? Conversely: fix
I would be very happy, if this is worked out somewhere, to be pointed to a number $r$, can one describe the non-semistable vector bundles on $P^2$ of with maximal destabilizing bundle of rank $r$?reference.
I can see the resemblance of my question to this other question on moduli spaces of vector bundles with fixed HN filtration factors, but I am hoping something more concrete is known, perhaps in the low rank situations. More precisely, I would be happy with a reference or an answer to the following: given a non-semistable vector bundles on $P^2$ of rank 4, under what conditions is the rank of the maximal destabilizing subbundle 3,2,1 respectively.