In mathematics, a Hirzebruch surface is a ruled surface over the projective line. They were studied by Friedrich Hirzebruch (1951).
Definition[edit]
The Hirzebruch surface is the -bundle, called a Projective bundle, over associated to the sheaf
GIT quotient[edit]
One method for constructing the Hirzebruch surface is by using a GIT quotient[1]: 21
Transition maps[edit]
One way to construct this -bundle is by using transition functions. Since affine vector bundles are necessarily trivial, over the charts of defined by there is the local model of the bundle
Properties[edit]
Projective rank 2 bundles over P1[edit]
Note that by Grothendieck's theorem, for any rank 2 vector bundle on there are numbers such that
Isomorphisms of Hirzebruch surfaces[edit]
In particular, the above observation gives an isomorphism between and since there is the isomorphism vector bundles
Analysis of associated symmetric algebra[edit]
Recall that projective bundles can be constructed using Relative Proj, which is formed from the graded sheaf of algebras
Intersection theory[edit]
Hirzebruch surfaces for n > 0 have a special rational curve C on them: The surface is the projective bundle of O(−n) and the curve C is the zero section. This curve has self-intersection number −n, and is the only irreducible curve with negative self intersection number. The only irreducible curves with zero self intersection number are the fibers of the Hirzebruch surface (considered as a fiber bundle over P1). The Picard group is generated by the curve C and one of the fibers, and these generators have intersection matrix
See also[edit]
References[edit]
- ^ a b c Manetti, Marco (2005-07-14). "Lectures on deformations of complex manifolds". arXiv:math/0507286.
- ^ Gathmann, Andreas. "Algebraic Geometry" (PDF). Fachbereich Mathematik - TU Kaiserslautern.
- ^ "Section 27.20 (02NB): Twisting by invertible sheaves and relative Proj—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-05-23.
- Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225
- Beauville, Arnaud (1996), Complex algebraic surfaces, London Mathematical Society Student Texts, vol. 34 (2nd ed.), Cambridge University Press, ISBN 978-0-521-49510-3, MR1406314
- Hirzebruch, Friedrich (1951), "Über eine Klasse von einfachzusammenhängenden komplexen Mannigfaltigkeiten", Mathematische Annalen, 124: 77–86, doi:10.1007/BF01343552, hdl:21.11116/0000-0004-3A56-B, ISSN 0025-5831, MR 0045384, S2CID 122844063