Simple Helices on Fano Threefolds

Abstract

AbstractBuilding on the work of Nogin, we prove that the braid groupB4acts transitively on full exceptional collections of vector bundles on Fano threefolds withb2= 1 andb3= 0. Equivalently, this group acts transitively on the set of simple helices (considered up to a shift in the derived category) on such a Fano threefold. We also prove that on threefolds withb2= 1 and very ample anticanonical class, every exceptional coherent sheaf is locally free.