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A conjecture of Hartshorne states that any vector bundle of a small rank on a projective space of large dimension is split (i.e. isomorphic to a direct sum of line bundles). I would like to know the current state of the art concerning this conjecture for vector bundles of rank $3$.

I work over a field $k$ (of characteristic $0$ if you prefer, but not necessarily algebraically closed).

What is the smallest value of $n_0$ known for which every vector bundle of rank $3$ on $\mathbb{P}_k^n$ is split for all $n > n_0$?

My knowledge on the subject is pretty poor. For example, I don't even know if $n_0$ is known to exist.

Much of the literature seems to be concerned with just rank $2$ vector bundles, and I was struggling to find anything about vector bundles of higher rank.

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    $\begingroup$ @Count Dracula: I read in the paper you mention: "I do not feel that I have sufficient evidence to formulate a conjecture about bundles of rank >2" (says Hartshorne). That doesn't seem to me sufficient ground to talk about the "Hartshorne conjecture" for rank >2 bundles. $\endgroup$
    – abx
    Commented Jan 17, 2017 at 14:30
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    $\begingroup$ I do not think whether the two questions mentioned above are known to be equivalent, except in the codimension 2 case and rank 2 vector bundles. Even in this case, one of the important ingredients is Barth's theorem, which is valid for all low codimensional varieties. For codimension 2, of course Serre's construction plays the deciding tool in one direction. There is also the published (Inv. Math.) incorrect proof that unstable bundles of rank 2 are split if $n\geq 4$ over complex numbers, due to Remmert and Schneider (I think). $\endgroup$
    – Mohan
    Commented Jan 17, 2017 at 14:36
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    $\begingroup$ @abx It is a time honored tradition in Algebraic Geometry to misattribute conjectures and I for one will continue to do so... $\endgroup$
    – Count Dracula
    Commented Jan 17, 2017 at 16:13
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    $\begingroup$ @CountDracula. In that case, you should offer us some questions that we can misattribute to you as conjectures :) $\endgroup$
    – Jason Starr
    Commented Jan 17, 2017 at 16:54
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    $\begingroup$ "On a conjecture of Count Dracula" - I'd like to see that on the arxiv. $\endgroup$
    – byu
    Commented Jan 19, 2017 at 17:37

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