Simple Helices on Fano Threefolds

http://dx.doi.org/10.4153/CMB-2010-106-x
Canad. Math. Bull. 54(2011), 520-526

  Published:2010-12-31
 Printed: Sep 2011

Building on the work of Nogin, we prove that the braid group $B_4$ acts transitively on full exceptional collections of vector bundles on Fano threefolds with $b_2=1$ and $b_3=0$. 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 with $b_2=1$ and very ample anticanonical class, every exceptional coherent sheaf is locally free.