Simple Helices on Fano Threefolds

Abstract 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.

[1]  R. Miró-Roig,et al.  THE STABILITY OF EXCEPTIONAL BUNDLES ON COMPLETE INTERSECTION 3-FOLDS , 2008 .

[2]  P. Pragacz.  Algebraic cycles, sheaves, shtukas, and moduli , 2008 .

[3]  P. Pragacz.  Algebraic cycles, sheaves, shtukas, and moduli : impanga lecture notes , 2008 .

[4]  A. Langer.  Lectures on Torsion-free Sheaves and Their Moduli , 2007 .

[5]  T. Bridgeland.  T-structures on some local Calabi–Yau varieties , 2005, math/0502050.

[6]  H. Uehara,et al.  Autoequivalences of derived categories on the minimal resolutions of An-singularities on surfaces , 2004, math/0409151.

[7]  S. Kuleshov.  Helix Theory , 2004 .

[8]  Richard P. Thomas,et al.  Braid group actions on derived categories of coherent sheaves , 2000, math/0001043.

[9]  D. Huybrechts,et al.  The geometry of moduli spaces of sheaves , 1997 .

[10]  L. Positselski.  All strictly exceptional collections in $D^b_{coh}(P^m)$ consist of vector bundles , 1995, alg-geom/9507014.

[11]  F A Bogomolov.  STABLE VECTOR BUNDLES ON PROJECTIVE SURFACES , 1995 .

[12]  F. Bogomolov.  STABLE VECTOR BUNDLES ON PROJECTIVE SURFACES , 1995 .

[13]  A. Bondal,et al.  HOMOLOGICAL PROPERTIES OF ASSOCIATIVE ALGEBRAS: THE METHOD OF HELICES , 1994 .

[14]  D. Nogin.  Helices on some Fano threefolds : constructivity of semiorthogonal bases of $K_0$ , 1994 .

[15]  A. Bondal,et al.  REPRESENTATION OF ASSOCIATIVE ALGEBRAS AND COHERENT SHEAVES , 1990 .

[16]  S. Kuleshov.  Helices and Vector Bundles: Seminaire Rudakov: Exceptional Bundles on K 3 Surfaces , 1990 .

[17]  S. Zube.  Helices and Vector Bundles: Seminaire Rudakov: Stability of Exceptional Bundles on Three Dimensional Projective Space , 1990 .

[18]  A. Beilinson.  Coherent sheaves on Pn and problems of linear algebra , 1978 .

[19]  B. G. Moĭšezon.  ALGEBRAIC HOMOLOGY CLASSES ON ALGEBRAIC VARIETIES , 1967 .

[20]  David Mumford,et al.  Lectures on curves on an algebraic surface , 1966 .