Horrocks–Mumford bundle explained

In algebraic geometry, the Horrocks–Mumford bundle is an indecomposable rank 2 vector bundle on 4-dimensional projective space P⁴ introduced by . 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.

\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.

See also

• List of algebraic surfaces

References

• • • Sasakura, Nobuo. Enta, Yoichi. Kagesawa, Masataka . Construction of rank two reflexive sheaves with similar properties to the Horrocks–Mumford bundle . Proc. Japan Acad., Ser. A . 69 . 5 . 144–148 . 1993 . 10.3792/pjaa.69.144. free .

• Projective geometry of elliptic curves - contains chapter on constructions of the bundle