### Impacto

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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Official URL: http://arxiv.org/pdf/math.AG/9804033.pdf

## Abstract

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

0617.14033)].

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 |
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Uncontrolled Keywords: | Cohen_Macaulay modules; hypersurface singularities |

Subjects: | Sciences > Mathematics > Algebraic geometry |

ID Code: | 14826 |

References: | [AG] E. Arrondo; B. Gra˜na; Vector bundles on G(1, 4) without intermediate cohomology; Preprint-1998. [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), 729-741. |

Deposited On: | 18 Apr 2012 09:26 |

Last Modified: | 06 Feb 2014 10:10 |

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