# Hirzebruch surface

In mathematics, a **Hirzebruch surface** is a ruled surface over the projective line. They were studied by Friedrich Hirzebruch (1951).

## Definition

The Hirzebruch surface
Σ_{n} is the **P**^{1} bundle over **P**^{1}
associated to the sheaf

The notation here means: O(*n*) is the *n*-th tensor power of the Serre twist sheaf O(1), the invertible sheaf or line bundle with associated Cartier divisor a single point. The surface Σ_{0} is isomorphic to **P**^{1}×**P**^{1}, and Σ_{1} is isomorphic to **P**^{2} blown up at a point so is not minimal.

## Properties

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 **P**^{1}). The Picard group is generated by the curve *C* and one of the fibers, and these generators have intersection matrix

so the bilinear form is two dimensional unimodular, and is even or odd depending on whether *n* is even or odd.

The Hirzebruch surface Σ_{n} (*n* > 1) blown up at a point on the special curve *C* is isomorphic to Σ_{n+1} blown up at a point not on the special curve.

## External links

### References

- 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.,**4**, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225 - Beauville, Arnaud (1996),
*Complex algebraic surfaces*, London Mathematical Society Student Texts,**34**(2nd ed.), Cambridge University Press, ISBN 978-0-521-49510-3, ISBN 978-0-521-49842-5 MR 1406314 - Hirzebruch, Friedrich (1951), "Über eine Klasse von einfachzusammenhängenden komplexen Mannigfaltigkeiten",
*Mathematische Annalen*,**124**: 77–86, doi:10.1007/BF01343552, ISSN 0025-5831, MR 0045384