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| author1-link=Geoffrey Horrocks
| author1-link=Geoffrey Horrocks
|last1=Horrocks|last2= G.
|last1=Horrocks|last2= G.
|author2-link=David Mumford|last2=Mumford|first2= D.]]
|author2-link=David Mumford|last2=Mumford|first2= D.
| title = A rank 2 vector bundle on ''P''<sup>4</sup> with 15000 symmetries
| title = A rank 2 vector bundle on ''P''<sup>4</sup> with 15000 symmetries
| journal = Topology
| journal = Topology

Revision as of 14:11, 7 November 2012

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 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.

See also

References

  • Horrocks; 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, MR0382279
  • Hulek, Klaus (1995), "The Horrocks–Mumford bundle", Vector bundles in algebraic geometry (Durham, 1993), London Math. Soc. Lecture Note Ser., vol. 208, Cambridge University Press, pp. 139–177, doi:10.1017/CBO9780511569319.007, MR1338416
  • 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)