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Construction of low rank vector bundles on \(\mathbb{P}^4\) and \(\mathbb{P}^5\). (English) Zbl 1076.14529

The authors describe a technique to obtain rank two vector bundles on special hypersurfaces of \(\mathbb{P}^5\), and apply this to give an explicit construction of rank two and rank three vector bundles on the projective spaces \(\mathbb{P}^4\), \(\mathbb{P}^5\), and the quadric \(S_5\) in \(\mathbb{P}^6\).
The starting point are so-called four generated rank two bundles. Let \(Y\subset X\) be a Cartier divisor on a projective space, and let \(B\) be a rank two vector bundle on \(Y\). Furthermore, assume that \(B\), as a coherent sheaf on \(X\), is the quotient of a sum of four line bundles \(\mathcal F\) on \(X\). The kernel \(\mathcal G\) of the map \(\mathcal F\to B\) is then used to find indecomposable rank three or rank two bundles \(\mathcal E\) on \(X\); namely, one tries to find line subbundles or line quotient bundles of \(\mathcal G\).
Explicit calculations are performed in the case when \(Y\subset \mathbb{P}^5\) is the homogeneous pfaffian of a \(4\times 4\) matrix. Here, the bundles can be described in terms of matrices. For several cases the authors show that the quotient \(\mathcal E\) of \(\mathcal G\) is an indecomposable rank three bundle. In positive characteristic, Frobenius pullbacks are used, while in characteristic zero the authors prove indecomposability in the case of reducible \(Y\subset \mathbb{P}^4\).
The construction of rank two bundles can be performed similarly by considering both a line subbundle and a line quotient bundle of \(\mathcal G\). This leads to the construction of indecomposable rank two bundles on \(\mathbb{P}^4\) and \(\mathbb{P}^5\) in characteristic 2, on \(\mathbb{P}^4\) in positive characteristic, and on the quadric \(S_5\) in \(\mathbb{P}^6\) in any characteristic.

MSC:

14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14N25 Varieties of low degree
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References:

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