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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 \Sigma_n is the \mathbb^1-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 \mathbb^1, 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 ...
\mathcal\oplus \mathcal(-n).The notation here means: \mathcal(n) is the -th tensor power of the Serre twist sheaf \mathcal(1), 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 \Sigma_0 is isomorphic to \mathbb P^1\times \mathbb P^1; and \Sigma_1 is isomorphic to the projective plane \mathbb P^2 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 ...
: \Sigma_n = (\Complex^2-\)\times (\Complex^2-\)/(\Complex^*\times\Complex^*) where the action of \Complex^*\times\Complex^* is given by (\lambda, \mu)\cdot(l_0,l_1,t_0,t_1) = (\lambda l_0, \lambda l_1, \mu t_0,\lambda^\mu t_1)\ . This action can be interpreted as the action of \lambda on the first two factors comes from the action of \Complex^* on \Complex^2 - \ defining \mathbb^1, and the second action is a combination of the construction of a direct sum of line bundles on \mathbb^1 and their projectivization. For the direct sum \mathcal\oplus \mathcal(-n) this can be given by the quotient variety\mathcal\oplus \mathcal(-n) = (\Complex^2-\)\times \Complex^2/\Complex^*where the action of \Complex^* is given by\lambda \cdot (l_0,l_1,t_0,t_1) = (\lambda l_0, \lambda l_1,\lambda^0 t_0=t_0, \lambda^ t_1)Then, the projectivization \mathbb(\mathcal\oplus\mathcal(-n)) is given by another \Complex^*-action sending an equivalence class _0,l_1,t_0,t_1\in\mathcal\oplus\mathcal(-n) to\mu \cdot _0,l_1,t_0,t_1= _0,l_1,\mu t_0,\mu t_1/math>Combining these two actions gives the original quotient up top.


Transition maps

One way to construct this \mathbb^1-bundle is by using transition functions. Since affine vector bundles are necessarily trivial, over the charts U_0,U_1 of \mathbb^1 defined by x_i \neq 0 there is the local model of the bundleU_i\times \mathbb^1Then, the transition maps, induced from the transition maps of \mathcal\oplus \mathcal(-n) give the mapU_0\times\mathbb^1, _ \to U_1\times\mathbb^1, _sending(X_0, _0:y_1 \mapsto (X_1, _0:x_0^n y_1where X_i is the affine coordinate function on U_i.


Properties


Projective rank 2 bundles over P1

Note that by Grothendieck's theorem, for any rank 2 vector bundle E on \mathbb P^1 there are numbers a,b \in \mathbb Z such thatE \cong \mathcal(a)\oplus \mathcal(b).As taking the projective bundle is invariant under tensoring by a line bundle, the ruled surface associated to E = \mathcal O(a) \oplus \mathcal O(b) is the Hirzebruch surface \Sigma_ since this bundle can be tensored by \mathcal(-a).


Isomorphisms of Hirzebruch surfaces

In particular, the above observation gives an isomorphism between \Sigma_n and \Sigma_ since there is the isomorphism vector bundles\mathcal(n)\otimes(\mathcal \oplus \mathcal(-n)) \cong \mathcal(n) \oplus \mathcal


Analysis of associated symmetric algebra

Recall that projective bundles can be constructed using Relative Proj, which is formed from the graded sheaf of algebras\bigoplus_^\infty \operatorname^i(\mathcal\oplus \mathcal(-n))The first few symmetric modules are special since there is a non-trivial anti-symmetric \operatorname^2-module \mathcal\otimes \mathcal(-n). These sheaves are summarized in the table\begin \operatorname^0(\mathcal\oplus \mathcal(-n)) &= \mathcal \\ \operatorname^1(\mathcal\oplus \mathcal(-n)) &= \mathcal \oplus \mathcal(-n) \\ \operatorname^2(\mathcal\oplus \mathcal(-n)) &= \mathcal \oplus \mathcal(-2n) \endFor i > 2 the symmetric sheaves are given by\begin \operatorname^k(\mathcal\oplus \mathcal(-n)) &= \bigoplus_^k \mathcal^\otimes \mathcal(-in) \\ &\cong \mathcal\oplus \mathcal(-n) \oplus \cdots \oplus \mathcal(-kn) \end


Intersection theory

Hirzebruch surfaces for have a special
rational curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane cu ...
on them: The surface is the projective bundle of \mathcal(-n) and the curve is the
zero section In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every po ...
. 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 \mathbb P^1). The
Picard group In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ver ...
is generated by the curve and one of the fibers, and these generators have intersection
matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...
\begin0 & 1 \\ 1 & -n \end , so 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.


Toric variety

The Hirzebruch surface \Sigma_n can be given an
action Action may refer to: * Action (philosophy), something which is done by a person * Action principles the heart of fundamental physics * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video gam ...
of the complex torus T = \mathbb^*\times \mathbb^*, with one \mathbb^* acting on the base \mathbb^1 with two fixed axis points, and the other \mathbb^* acting on the fibers of the vector bundle \mathcal\oplus \mathcal(-n), specifically on the first line bundle component, and hence on the projective bundle. This produces an open orbit of ''T'', making \Sigma_n a
toric variety In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be ...
. Its associated fan partitions the standard lattice \mathbb^2 into four cones (each corresponding to a coordinate chart), separated by the rays along the four vectors:
(1,0), (0,1), (0,-1), (-1,n).
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. Any smooth toric surface except \mathbb^2 can be constructed by repeatedly
blowing up In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with the space of all directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the poin ...
a Hirzebruch surface at ''T''-fixed points.


See also

*
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 ...


References

* * *{{Citation, last1=Hirzebruch , first1=Friedrich , author1-link=Friedrich Hirzebruch , title=Über eine Klasse von einfachzusammenhängenden komplexen Mannigfaltigkeiten , doi=10.1007/BF01343552 , mr=0045384 , year=1951 , journal=
Mathematische Annalen ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...
, issn=0025-5831 , volume=124 , pages=77–86, hdl=21.11116/0000-0004-3A56-B , s2cid=122844063 , hdl-access=free


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