RT Journal Article
T1 Vector bundles on fano 3-folds without intermediate cohomology
A1 Arrondo Esteban, Enrique
A1 Costa, Laura
AB 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 intermediatecohomology if and only if it decomposes as a direct sum of line bundles. It is also knownthat 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; Zbl0617.14033)].In the paper under review the authors deal with vector bundles without intermediatecohomology on some Fano 3-folds with second Betti number b2 = 1. The Fano 3-foldsthey consider are smooth cubics in P4, smooth complete intersection of type (2, 2) in P5and smooth 3-dimensional linear sections of G(1, 4)  P9. A complete classification ofrank two vector bundles without intermediate cohomology on such 3-folds is given. Infact the authors prove that, up to a twist, there are only three indecomposable vectorbundles without intermediate cohomology. Vector bundles of rank greater than two arealso considered. Under an additional technical condition, the authors characterize thepossible Chern classes of such vector bundles without intermediate cohomology.
PB Marcel Dekker
SN 0092-7872
YR 2000
FD 2000
LK https://hdl.handle.net/20.500.14352/57164
UL https://hdl.handle.net/20.500.14352/57164
LA eng
NO [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 onhypersurface singularities II, Invent. Math. 88 (1987), 165-182.[Ho] G. Horrocks; Vector bundles on the punctured spectrum of a ring, Proc. LondonMath. 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 inP4, to appear in Rendiconti di Torino.[O1] G. Ottaviani; Crit`eres de scindage pour les fibr´es vectoriels sur les grassmannianeset 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 Grassmanniansand quadrics, Annali Mat. Pura Appl. (IV) 155 (1989), 317-341.[SW] M. Szurek; J.A. Wi´sniewski; Conics, conic fibrations and stable vector bundlesof rank 2 on some Fano threefolds, Rev. Roumaine Math. Pures Appl. 38 (1993),729-741.
NO DGICYT
DS Docta Complutense
RD 7 may 2024