Simple helices on Fano threefolds

TL;DR

This paper demonstrates the transitive action of the braid group on full exceptional collections on certain Fano threefolds and shows that all exceptional sheaves are locally free under specific conditions.

Contribution

It proves the braid group acts transitively on exceptional collections on Fano threefolds with specific Betti numbers and establishes local freeness of exceptional sheaves in certain cases.

Findings

Braid group $B_4$ acts transitively on exceptional collections.

Exceptional sheaves are locally free on Fano threefolds with $b_2=1$ and very ample anticanonical class.

Abstract

Building on the work of Nogin \cite{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.