Hirzebruch surface


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

Definition

The Hirzebruch surface is the -bundle, called a Projective bundle, over associated to the sheaf
The notation here means: is the n-th tensor power of the Serre twist sheaf, the invertible sheaf or line bundle with associated Cartier divisor a single point. The surface is isomorphic to P1 × P1, and is isomorphic to P2 blown up at a point so is not minimal.

GIT quotient

One method for constructing the Hirzebruch surface is by using a GIT quotientpg 21
where the action of is given by
This action can be interpreted as the action of on the first two factors comes from the action of on defining, and the second action is a combination of the construction of a direct sum of line bundles on and their projectivization. For the direct sum this can be given by the quotient varietypg 24
where the action of is given by
Then, the projectivization is given by another -actionpg 22 sending an equivalence class to
Combining these two actions gives the original quotient up top.

Transition maps

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
Then, the transition maps, induced from the transition maps of give the map
sending
where is the affine coordinate function on.

Properties

Projective rank 2 bundles over P1

Note that the projective bundle
is equivalent to a Hirzebruch surface since projective bundles are invariant after tensoring by a line bundle. In particular, this is associated to the Hirzebruch surface since this bundle can be tensored by the line bundle.

Isomorphisms of Hirzebruch surfaces

In particular, the above observation gives an isomorphism between and since there is the isomorphism vector bundles

Analysis of associated symmetric algebra

Recall that projective bundles can be constructed using Relative Proj, which is formed from the graded sheaf of algebras
The first few symmetric modules are special since there is a non-trivial anti-symmetric -module. These sheaves are summarized in the table
For the symmetric sheaves are given by

Properties

Hirzebruch surfaces for n > 0 have a special rational curve C on them: The surface is the projective bundle of O 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. 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 Σn 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.