Simple Helices on Fano Threefolds

DOI10.4153/CMB-2010-106-XzbMATH Open1228.14014arXiv0710.4195OpenAlexW2963301265MaRDI QIDQ3095123

Alexander Polishchuk

Publication date: 28 October 2011

Published in: Canadian Mathematical Bulletin (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/0710.4195

Mathematics Subject Classification ID

Vector bundles on surfaces and higher-dimensional varieties, and their moduli (14J60) Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) (46M20) Fano varieties (14J45) Vector bundles on curves and their moduli (14H60) Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) (18F20) Holomorphic fiber spaces (32Lxx)

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