In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the Horrocks–Mumford bundle is an indecomposable rank 2
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
on 4-dimensional
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
''P''
4 introduced by . It is the only such bundle known, although a generalized construction involving
Paley graph
In mathematics, Paley graphs are dense undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, whic ...
s produces other rank 2
sheaves (Sasukara et al. 1993). The zero sets of sections of the Horrocks–Mumford bundle are
abelian surface In mathematics, an abelian surface is a 2-dimensional abelian variety.
One-dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic via the Riemann bi ...
s of degree 10, called Horrocks–Mumford surfaces.
By computing
Chern classes
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ma ...
one sees that the second
exterior power
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
of the Horrocks–Mumford bundle ''F'' is the line bundle ''O(5)'' on ''P
4''. Therefore, the zero set ''V'' of a general section of this bundle is a
quintic threefold In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space \mathbb^4. Non-singular quintic threefolds are Calabi–Yau manifolds.
The Hodge diamond of a non-singular quintic 3-fold is
Mathem ...
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
This is a list of named algebraic surfaces, compact complex surfaces, and families thereof, sorted according to their Kodaira dimension following Enriques–Kodaira classification.
Kodaira dimension −∞
Rational surfaces
* Projective plane Qu ...
References
*
*
*
Algebraic varieties
Vector bundles
Projective geometry of elliptic curves- contains chapter on constructions of the bundle
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