They were studied by Friedrich Hirzebruch (1951).
is the n-th tensor power of the Serre twist sheaf
, the invertible sheaf or line bundle with associated Cartier divisor a single point.
is isomorphic to the projective plane
One method for constructing the Hirzebruch surface is by using a GIT quotient[1]: 21 :
, and the second action is a combination of the construction of a direct sum of line bundles on
-action[1]: 22 sending an equivalence class
Combining these two actions gives the original quotient up top.
Since affine vector bundles are necessarily trivial, over the charts
there is the local model of the bundle
is the affine coordinate function on
[2] Note that by Grothendieck's theorem, for any rank 2 vector bundle
As taking the projective bundle is invariant under tensoring by a line bundle,[3] the ruled surface associated to
In particular, the above observation gives an isomorphism between
since there is the isomorphism vector bundles
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
These sheaves are summarized in the table
Hirzebruch surfaces for n > 0 have a special rational curve C on them: The surface is the projective bundle of
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
The Picard group is generated by the curve C and one of the fibers, and these generators have intersection matrix
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.
can be given an action of the complex torus
acting on the fibers of the vector bundle
This produces an open orbit of T, making
Its associated fan partitions the standard lattice
into four cones (each corresponding to a coordinate chart), separated by the rays along the four vectors:[4]
All the theory above generalizes to arbitrary toric varieties, including the construction of the variety as a quotient and by coordinate charts, as well as the explicit intersection theory.
can be constructed by repeatedly blowing up a Hirzebruch surface at T-fixed points.