Horrocks–Mumford bundle: Difference between revisions

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In algebraic geometry, the Horrocks–Mumford bundle is an indecomposable rank 2 vector bundle on 4-dimensional projective space P⁴ introduced by Geoffrey Horrocks and David Mumford (1973). It is the only such bundle known, although a generalized construction involving Paley graphs produces other rank 2 sheaves (Sasukara et al. 1993). The zero sets of sections of the Horrocks–Mumford bundle are abelian surfaces of degree 10, called Horrocks–Mumford surfaces.

By computing Chern classes one sees that the second exterior power $\wedge ^{2}F$ of the Horrocks–Mumford bundle F is the line bundle O(5) on P⁴. 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.

References

Horrocks, G.; Mumford, D. (1973). "A rank 2 vector bundle on P⁴ with 15000 symmetries". Topology. 12: 63–81. doi:10.1016/0040-9383(73)90022-0.{{cite journal}}: CS1 maint: multiple names: authors list (link)

Sasakura, Nobuo; Enta, Yoichi; Kagesawa, Masataka (1993). "Construction of rank two reflexive sheaves with similar properties to the Horrocks–Mumford bundle". Proc. Japan Acad., Ser. A. 69 (5): 144–148. doi:10.3792/pjaa.69.144.{{cite journal}}: CS1 maint: multiple names: authors list (link)

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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'''.
-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'''.
-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.
+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.
 ==See also==