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Rank 4 vector bundles on the quintic threefold. (English) Zbl 1106.14029

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.

MSC:

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14J30 \(3\)-folds

Citations:

Zbl 1165.14304

Software:

schubert
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Full Text: DOI arXiv

References:

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