In mathematics, a Hirzebruch surface is a
ruled surface
In geometry, a Differential geometry of surfaces, surface in 3-dimensional Euclidean space is ruled (also called a scroll) if through every Point (geometry), point of , there is a straight line that lies on . Examples include the plane (mathemat ...
over the
projective line
In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a '' point at infinity''. The statement and the proof of many theorems of geometry are simplified by the ...
. They were studied by .
Definition
The Hirzebruch surface
is the
-bundle (a
projective bundle
In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces.
By definition, a scheme ''X'' over a Noetherian scheme ''S'' is a P''n''-bundle if it is locally a projective ''n''-space; i.e., X \times_S U \simeq \mathbb ...
) over the projective line
, associated to the
sheaf
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics)
In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open s ...
The notation here means:
is the -th tensor power of the
Serre twist sheaf , the
invertible sheaf
In mathematics, an invertible sheaf is a sheaf on a ringed space that has an inverse with respect to tensor product of sheaves of modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle. Due to their intera ...
or
line bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organis ...
with associated
Cartier divisor
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumf ...
a single point. The surface
is isomorphic to
; and
is isomorphic to the projective plane
blown up at a point, so it is not minimal.
GIT quotient
One method for constructing the Hirzebruch surface is by using a
GIT quotient In algebraic geometry, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme X = \operatorname A with an action by a group scheme ''G'' is the affine scheme \operatorname(A^G), the prime spectrum of the ring of ...
:
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 variety
where the action of
is given by
Then, the projectivization
is given by another
-action
sending an equivalence class
to