# Rank 4 vector bundles on the quintic threefold

Open Mathematics (2005)

- Volume: 3, Issue: 3, page 404-411
- ISSN: 2391-5455

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topCarlo Madonna. "Rank 4 vector bundles on the quintic threefold." Open Mathematics 3.3 (2005): 404-411. <http://eudml.org/doc/268833>.

@article{CarloMadonna2005,

abstract = {By the results of the author and Chiantini in [3], on a general quintic threefold X⊂P 4 the minimum integer p for which there exists a positive dimensional family of irreducible rank p vector bundles on X without intermediate cohomology is at least three. In this paper we show that p≤4, by constructing series of positive dimensional families of rank 4 vector bundles on X without intermediate cohomology. The general member of such family is an indecomposable bundle from the extension class Ext 1 (E, F), for a suitable choice of the rank 2 ACM bundles E and F on X. The existence of such bundles of rank p=3 remains under question.},

author = {Carlo Madonna},

journal = {Open Mathematics},

keywords = {14J60},

language = {eng},

number = {3},

pages = {404-411},

title = {Rank 4 vector bundles on the quintic threefold},

url = {http://eudml.org/doc/268833},

volume = {3},

year = {2005},

}

TY - JOUR

AU - Carlo Madonna

TI - Rank 4 vector bundles on the quintic threefold

JO - Open Mathematics

PY - 2005

VL - 3

IS - 3

SP - 404

EP - 411

AB - By the results of the author and Chiantini in [3], on a general quintic threefold X⊂P 4 the minimum integer p for which there exists a positive dimensional family of irreducible rank p vector bundles on X without intermediate cohomology is at least three. In this paper we show that p≤4, by constructing series of positive dimensional families of rank 4 vector bundles on X without intermediate cohomology. The general member of such family is an indecomposable bundle from the extension class Ext 1 (E, F), for a suitable choice of the rank 2 ACM bundles E and F on X. The existence of such bundles of rank p=3 remains under question.

LA - eng

KW - 14J60

UR - http://eudml.org/doc/268833

ER -

## References

top- [1] E. Arrondo and L. Costa: “Vector bundles on Fano 3-folds without intermediate cohomology”, Comm. Algebra, Vol. 28, (2000), pp. 3899–3911. Zbl1004.14010
- [2] R.O. Buchweitz, G.M. Greuel and F.O. Schreyer: “Cohen-Macaulay modules on hypersurface singularities II”, Invent. Math., Vol. 88, (1987), pp. 165–182. http://dx.doi.org/10.1007/BF01405096 Zbl0617.14034
- [3] L. Chiantini and C. Madonna: “ACM bundles on a general quintic threefold”, Matematiche (Catania), Vol. 55, (2000), pp. 239–258. Zbl1165.14304
- [4] R. Hartshorne: “Stable vector bundles of rank 2 on ℙ3”, Math. Ann., Vol. 238, (1978), pp. 229–280. http://dx.doi.org/10.1007/BF01420250 Zbl0411.14002
- [5] S. Katz and S. Stromme: Schubert, a Maple package for intersection theory and enumerative geometry, from website http://www.mi.uib.no/schubert/
- [6] C.G. Madonna: “ACM bundles on prime Fano threefolds and complete intersection Calabi Yau threefolds”, Rev. Roumaine Math. Pures Appl., Vol. 47, (2002), pp. 211–222. Zbl1051.14050
- [7] C. Okonek, M. Schneider and H. Spindler: “Vector bundles on complex projective spaces”, Progress in Mathematics, Vol. 3, (1980), pp. 389. Zbl0438.32016

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