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| In [[algebraic geometry]], the '''Horrocks–Mumford bundle''' is an indecomposable rank 2 [[vector bundle]] on 4-dimensional [[projective space]] ''P''<sup>4</sup> introduced by {{harvs|txt|author1-link= Geoffrey Horrocks |author2-link=David Mumford|first=Geoffrey|last= Horrocks |first2=David|last2= Mumford|year=1973}}. It is the only such bundle known, although a generalized construction involving [[Paley graph]]s produces other rank 2 [[Sheaf (mathematics)|sheaves]] (Sasukara et al. 1993). The zero sets of sections of the Horrocks–Mumford bundle are [[abelian surface]]s of degree 10, called '''Horrocks–Mumford surfaces'''.
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| By computing [[Chern classes]] one sees that the second [[exterior power]] <math> \wedge^2 F </math> of the Horrocks–Mumford bundle ''F'' is the line bundle ''O(5)'' on ''P<sup>4</sup>''. Therefore the zero set ''V'' of a general section of this bundle is a [[quintic threefold]] called a '''Horrocks–Mumford quintic'''. Such a ''V'' has exactly 100 nodes; there exists a small resolution ''V′'' which is a [[Calabi–Yau]] threefold fibered by Horrocks–Mumford surfaces.
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| ==See also==
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| *[[List of algebraic surfaces]]
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| ==References==
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| *{{citation
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| | author1-link=Geoffrey Horrocks
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| |last1=Horrocks|last2= G.
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| |author2-link=David Mumford|last2=Mumford|first2= D.
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| | title = A rank 2 vector bundle on ''P''<sup>4</sup> with 15000 symmetries
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| | journal = Topology
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| | volume = 12
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| | pages = 63–81
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| | year = 1973
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| | doi = 10.1016/0040-9383(73)90022-0
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| |id={{MR|0382279}}
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| }}
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| *{{Citation | last1=Hulek | first1=Klaus | title=Vector bundles in algebraic geometry (Durham, 1993) | publisher=[[Cambridge University Press]] | series=London Math. Soc. Lecture Note Ser. | doi=10.1017/CBO9780511569319.007 | id={{MathSciNet | id = 1338416}} | year=1995 | volume=208 | chapter=The Horrocks–Mumford bundle | pages=139–177}}
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| *{{cite journal
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| | author = Sasakura, Nobuo; Enta, Yoichi; Kagesawa, Masataka
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| | title = Construction of rank two reflexive sheaves with similar properties to the Horrocks–Mumford bundle
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| | journal = Proc. Japan Acad., Ser. A
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| | volume = 69
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| | issue = 5
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| | pages = 144–148
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| | year = 1993
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| | doi = 10.3792/pjaa.69.144}}
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| {{DEFAULTSORT:Horrocks-Mumford bundle}}
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| [[Category:Algebraic varieties]]
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| [[Category:Vector bundles]]
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| {{geometry-stub}}
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