Math gifts

- Art Gallery -


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


The Hirzebruch surface Σn is the P1 bundle over P1 associated to the sheaf

\( O(0)+O(-n).\ \)

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 Σ0 is isomorphic to P1×P1, and Σ1 is isomorphic to P2 blown up at a point so is not minimal.


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

\( \begin{bmatrix}0 & 1 \\ 1 & -n \end{bmatrix} , \)

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

The Hirzebruch surface Σn (n > 1) blown up at a point on the special curve C is isomorphic to Σn-1 blown up at a point not on the special curve.

External links

Manifold Atlas


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

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

Retrieved from ""
All text is available under the terms of the GNU Free Documentation License

Home - Hellenica World