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Vector bundles on fano 3-folds without intermediate cohomology


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

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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.

Item Type:Article
Uncontrolled Keywords:Cohen_Macaulay modules; hypersurface singularities
Subjects:Sciences > Mathematics > Algebraic geometry
ID Code:14826

[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),


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