Arrondo Esteban, Enrique and Costa, Laura (2000) Vector bundles on fano 3-folds without intermediate cohomology. Communications in Algebra, 28 (8). pp. 3899-3911. ISSN 0092-7872
Restricted to Repository staff only until 2020.
Official URL: http://arxiv.org/pdf/math.AG/9804033.pdf
A well known result of G. Horrocks [Proc. Lond. Math. Soc. (3) 14, 689-713 (1964;
Zbl 0126.16801)] says that a vector bundle on a projective space has no intermediate
cohomology if and only if it decomposes as a direct sum of line bundles. It is also known
that only on projective spaces and quadrics there is, up to a twist by a line bundle,
a finite number of indecomposable vector bundles with no intermediate cohomology
[see R.-O. Buchweitz, G.-M. Greuel and F.-O. Schreyer, Invent. Math. 88, 165-182
(1987; Zbl 0617.14034) and also H. Kn¨orrer, Invent. Math. 88, 153-164 (1987; Zbl
In the paper under review the authors deal with vector bundles without intermediate
cohomology on some Fano 3-folds with second Betti number b2 = 1. The Fano 3-folds
they consider are smooth cubics in P4, smooth complete intersection of type (2, 2) in P5
and smooth 3-dimensional linear sections of G(1, 4) P9. A complete classification of
rank two vector bundles without intermediate cohomology on such 3-folds is given. In
fact the authors prove that, up to a twist, there are only three indecomposable vector
bundles without intermediate cohomology. Vector bundles of rank greater than two are
also considered. Under an additional technical condition, the authors characterize the
possible Chern classes of such vector bundles without intermediate cohomology.
|Uncontrolled Keywords:||Cohen_Macaulay modules; hypersurface singularities|
|Subjects:||Sciences > Mathematics > Algebraic geometry|
[AG] E. Arrondo; B. Gra˜na; Vector bundles on G(1, 4) without intermediate cohomology;
[AS] E. Arrondo; I. Sols; On congruences of lines in the projective space; M´em. Soc.
Math. France 50 (1992).
[BGS] R.O. Buchweitz; G.M. Greuel; F.O. Schreyer; Cohen-Macaulay modules on
hypersurface singularities II, Invent. Math. 88 (1987), 165-182.
[Ho] G. Horrocks; Vector bundles on the punctured spectrum of a ring, Proc. London
Math. Soc. (3) 14 (1964), 689-713.
[Is] V.A. Iskovskih; Fano 3-Folds, I, Math. USSR Izvestija 11 (1977), 485-527.
[Kn] H. Kn¨orrer;Cohen-Macaulay modules on hypersurface singularities I, Invent.
Math. 88 (1987), 153-164.
[Ma] C. Madonna; A splitting criterion for rank 2 vector bundles on hypersurfaces in
P4, to appear in Rendiconti di Torino.
[O1] G. Ottaviani; Crit`eres de scindage pour les fibr´es vectoriels sur les grassmannianes
et les quadriques, C.R. Acad. Sci. Paris, t. 305, S´erie I (1987), 257-260.
[O2] G. Ottaviani; Some extensions of Horrocks criterion to vector bundles on Grassmannians
and quadrics, Annali Mat. Pura Appl. (IV) 155 (1989), 317-341.
[SW] M. Szurek; J.A. Wi´sniewski; Conics, conic fibrations and stable vector bundles
of rank 2 on some Fano threefolds, Rev. Roumaine Math. Pures Appl. 38 (1993),
|Deposited On:||18 Apr 2012 09:26|
|Last Modified:||06 Feb 2014 10:10|
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