I would like to know if there are some interesting partial orders defined on the isomorphism classes of vector bundles on $\mathbb P^n_k$ (you can assume $k$ is $\mathbb C$ if that helps).

Motivation: I am looking at a function which is related to a partial order on vector bundles on the punctured spectrum of a regular local ring. Since the obvious, and much more studied, analogue is vector bundles on $\mathbb P^n_k$, I am curious on what is known in that case. In the case $n=1$, since all vector bundles splits as direct sum of $\mathcal O(a)$, one can obviously use the set of twists to define a partial order. Thanks.

  • $\begingroup$ Stability would give you a natural order ($E < E'$ if and only if every HN-filtration factor of $E$ has smaller slope than every HN-filtration factors of $E'$), but it's hard to guess whether that is something that would be useful for your motivation. $\endgroup$ – Arend Bayer Sep 30 '10 at 19:04
  • $\begingroup$ Hi Arend, thanks, that's a good one. Do you know if this p.o. was actually used somewhere? And if you do, would you mind putting it in an answer, please? $\endgroup$ – Hailong Dao Sep 30 '10 at 19:32
  • $\begingroup$ @AByer: wait, did you change your username? Was that inspired by BCnrd (-: ? $\endgroup$ – Hailong Dao Sep 30 '10 at 20:31

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