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 sheafThe notation here means:  is the -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 , and  is isomorphic to  blown up at a point so is not minimal. GIT quotient 
One method for constructing the Hirzebruch surface is by using a GIT quotientwhere the action of  is given byThis 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 varietywhere the action of  is given byThen, the projectivization  is given by another -action sending an equivalence class  toCombining 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 bundleThen, the transition maps, induced from the transition maps of  give the mapsendingwhere  is the affine coordinate function on . Properties

Projective rank 2 bundles over P1 
Note that by Grothendieck's theorem, for any rank 2 vector bundle  on  there are numbers  such thatAs taking the projective bundle is invariant under tensoring by a line bundle, the ruled surface associated to  is the Hirzebruch surface  since this bundle can be tensored by . 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 algebrasThe first few symmetric modules are special since there is a non-trivial anti-symmetric -module . These sheaves are summarized in the tableFor  the symmetric sheaves are given by

Intersection theory 
Hirzebruch surfaces for  have a special rational curve  on them: The surface is the projective bundle of  and the curve  is the zero section. This curve has self-intersection number , 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 ). The Picard group is generated by the curve  and one of the fibers, and these generators have intersection matrixso the bilinear form is two dimensional unimodular, and is even or odd depending on whether  is even or odd. The Hirzebruch surface  () blown up at a point on the special curve  is isomorphic to  blown up at a point not on the special curve. See also 

 Projective bundle

References

External links 
 Manifold Atlas
https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/alggeom-2002-c10.pdf
https://mathoverflow.net/q/122952

Algebraic surfaces
Complex surfaces