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In 1978 Hartshorne published a list of 26 open problems about algebraic bundles on projective spaces [Hart], proceeding from an Oxford conference organized by Atiyah.

I understand that many of these problems now have well known solutions, but others are less familiar to me. Might anybody know which have solutions? I would love to read them.

It would also be nice to know which problems remain open.

[Hart] Hartshorne, Robin, Algebraic vector bundles on projective spaces: A problem list, Topology 18, 117-128 (1979). ZBL0417.14011.

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    $\begingroup$ The first few problems in the list are still open. We still do not know whether there are indecomposable rank 2 vector bundles on projective space of dimension at least 5. They exist in positive characteristic. The proof of Grauert and Schneider which he quotes (published in Inventiones) is generally believed to be wrong, so that too is open. In other words, are there unstable bundles of rank 2 in $\mathbb{P}^5$? Which ones do you think are true among those problems? $\endgroup$
    – Mohan
    Commented Dec 21, 2017 at 1:07
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    $\begingroup$ Problems four, six, and eight specifically. $\endgroup$
    – user100272
    Commented Dec 21, 2017 at 1:48
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    $\begingroup$ Yes, some of the moduli questions are understood. Most others are not known. $\endgroup$
    – Mohan
    Commented Dec 21, 2017 at 1:52

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