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 Σn is the P1 bundle over P1 associated to the sheaf

${\displaystyle O(0)+O(-n).\ }$

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.

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 numbern, 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

${\displaystyle {\begin{bmatrix}0&1\\1&-n\end{bmatrix}},}$

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.