A celebrated part in the classification of Fano threefold is Mukai's vector bundle method.
One of the main result is an existence (and rigidity) result for vector bundles with rank dividing the genus, for prime Fano threefolds of index 1.
Unfortunately, in the literature, the proof has a gap.
I will present joint work with Arend Bayer and Alexander Kuznetsov, where we fill this gap.