Madonna, Carlo Rank 4 vector bundles on the quintic threefold. (English) Zbl 1106.14029 Cent. Eur. J. Math. 3, No. 3, 404-411 (2005). Summary: By the results of the author and L. Chiantini [Matematiche 55, No. 2, 239–258 (2000; Zbl 1165.14304)], on a general quintic threefold \(X\subset \mathbb{P}^4\) 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 \(p\leq 4\), 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 \(\text{Ext}^1(E, F)\), 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. Cited in 3 Documents MSC: 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14J30 \(3\)-folds Citations:Zbl 1165.14304 Software:schubert PDFBibTeX XMLCite \textit{C. Madonna}, Cent. Eur. J. Math. 3, No. 3, 404--411 (2005; Zbl 1106.14029) Full Text: DOI arXiv References: [1] E. Arrondo and L. Costa: “Vector bundles on Fano 3-folds without intermediate cohomology”, Comm. Algebra, Vol. 28, (2000), pp. 3899-3911.; · Zbl 1004.14010 [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; · Zbl 0617.14034 [3] L. Chiantini and C. Madonna: “ACM bundles on a general quintic threefold”, Matematiche (Catania), Vol. 55, (2000), pp. 239-258.; · Zbl 1165.14304 [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; · Zbl 0411.14002 [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.; · Zbl 1051.14050 [7] C. Okonek, M. Schneider and H. Spindler: “Vector bundles on complex projective spaces”, Progress in Mathematics, Vol. 3, (1980), pp. 389.; · Zbl 0438.32016 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.