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Monads and cohomology modules of rank 2 vector bundles. (English) Zbl 0722.14005

Let be \(S={\mathbb{C}}[Z_ 1,...,Z_ n]\); for a vector bundle E on \({\mathbb{P}}_ n\), \(H^ i(E(*))=\oplus_{m}H^ i(E(m)) \) is a graded S- module. Let N be an arbitrary graded S-module of finite length. For \(n=2\), P. Rao [J. Algebra 86, 23-34 (1984; Zbl 0528.14008)] gives a necessary and sufficient condition for \(N\approx H^ 1(E(*))\) for some rank two vector bundle E, the author gives a set of necessary and sufficient conditions in case \(n=3\). If \(n\geq 4\), the only indecomposable rank 2 vector bundle known is the Horrocks-Mumford bundle F on \({\mathbb{P}}_ 4\). The author gives minimal free resolutions for \(H^ 2(F(*))\), \(H^ 1(F(*))\) and F using (the display of) the Horrocks- Mumford monad. He computes explicitly bases for the sections of F(m) and in particular obtains equations of zero schemes of sections of F(3), including abelian surfaces in \({\mathbb{P}}_ 4\).

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
18G25 Relative homological algebra, projective classes (category-theoretic aspects)

Citations:

Zbl 0528.14008

Software:

Macaulay2
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References:

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