Hirzebruch surface explained

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

Definition

The Hirzebruch surface

\Sigma _{n} is the P ^{1} -bundle (a projective bundle) over the projective line P ^{1} , associated to the sheaf$$\backslash mathcal\backslash oplus\; \backslash mathcal(-n).$$The notation here means: l{O}(n) is the -th tensor power of the Serre twist sheaf l{O}(1) , the invertible sheaf or line bundle with associated Cartier divisor a single point. The surface \Sigma _{0} is isomorphic to P ^{1 x} P ^{1} ; and \Sigma _{1} is isomorphic to the projective plane 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^[1] $$\backslash Sigma\_n\; =\; (\backslash Complex^2-\backslash )\backslash times\; (\backslash Complex^2-\backslash )/(\backslash Complex^*\backslash times\backslash Complex^*)$$where the action of

\Complex ^{* x \Complex} ^{*} is given by$$(\backslash lambda,\; \backslash mu)\backslash cdot(l\_0,l\_1,t\_0,t\_1)\; =\; (\backslash lambda\; l\_0,\; \backslash lambda\; l\_1,\; \backslash mu\; t\_0,\backslash lambda^\backslash mu\; t\_1)$$This action can be interpreted as the action of λ on the first two factors comes from the action of \Complex ^{*} on \Complex ^{2} - \{ 0 \} defining P ^{1} , and the second action is a combination of the construction of a direct sum of line bundles on P ^{1} and their projectivization. For the direct sum l{O} ⊕ l{O}(-n) this can be given by the quotient variety$$\backslash mathcal\backslash oplus\; \backslash mathcal(-n)\; =\; (\backslash Complex^2-\backslash )\backslash times\; \backslash Complex^2/\backslash Complex^*$$where the action of \Complex ^{*} is given by$$\backslash lambda\; \backslash cdot\; (l\_0,l\_1,t\_0,t\_1)\; =\; (\backslash lambda\; l\_0,\; \backslash lambda\; l\_1,\backslash lambda^a\; t\_0,\; \backslash lambda^0\; t\_1\; =\; t\_1)$$Then, the projectivization P ( l{O} ⊕ l{O}(-n)) is given by another \Complex ^{*} -action sending an equivalence class [l _{0,l} _{1,t} _{0,t} _{1]} \in l{O} ⊕ l{O}(-n) to$$\backslash mu\; \backslash cdot\; [l\_0,l\_1,t\_0,t\_1]\; =\; [l\_0,l\_1,\backslash mu\; t\_0,\backslash mu\; t\_1]$$Combining these two actions gives the original quotient up top.

Transition maps

One way to construct this

P ^{1} -bundle is by using transition functions. Since affine vector bundles are necessarily trivial, over the charts U _{0,U} _{1} of P ^{1} defined by x _{i} ≠ 0 there is the local model of the bundle$$U\_i\backslash times\; \backslash mathbb^1$$Then, the transition maps, induced from the transition maps of l{O} ⊕ l{O}(-n) give the map$$U\_0\backslash times\backslash mathbb^1|\_\; \backslash to\; U\_1\backslash times\backslash mathbb^1|\_$$sending$$(X\_0,\; [y\_0:y\_1])\; \backslash mapsto\; (X\_1,\; [y\_0:x\_0^n\; y\_1])$$where X _{i} is the affine coordinate function on U _{i} .^[2]

Properties

Projective rank 2 bundles over P¹

Note that by Grothendieck's theorem, for any rank 2 vector bundle

E on P ^{1} there are numbers a,b \in Z such that$$E\; \backslash cong\; \backslash mathcal(a)\backslash oplus\; \backslash mathcal(b).$$As taking the projective bundle is invariant under tensoring by a line bundle,^[3] the ruled surface associated to E = l O(a) ⊕ l O(b) is the Hirzebruch surface \Sigma _{b-a} since this bundle can be tensored by l{O}(-a) .

Isomorphisms of Hirzebruch surfaces

In particular, the above observation gives an isomorphism between

\Sigma _{n} and \Sigma _{-n} since there is the isomorphism vector bundles$$\backslash mathcal(n)\backslash otimes(\backslash mathcal\; \backslash oplus\; \backslash mathcal(-n))\; \backslash cong\; \backslash mathcal(n)\; \backslash oplus\; \backslash mathcal$$

Analysis of associated symmetric algebra

Recall that projective bundles can be constructed using Relative Proj, which is formed from the graded sheaf of algebras$$\backslash bigoplus\_^\backslash infty\; \backslash operatorname^i(\backslash mathcal\backslash oplus\; \backslash mathcal(-n))$$The first few symmetric modules are special since there is a non-trivial anti-symmetric

\operatorname{Alt} ^{2} -module l{O} ⊗ l{O}(-n) . These sheaves are summarized in the tableFor i > 2 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

l{O}(-n) 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 P ^{1} ). 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.

Toric variety

The Hirzebruch surface

\Sigma _{n} can be given an action of the complex torus T = C ^{* x} C ^{*} , with one C ^{*} acting on the base P ^{1} with two fixed axis points, and the other C ^{*} acting on the fibers of the vector bundle $\backslash mathcal\backslash oplus\; \backslash mathcal(-n)$, 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. Its associated fan partitions the standard lattice Z ^{2} into four cones (each corresponding to a coordinate chart), separated by the rays along the four vectors:^[4]

( 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

P ^{2} can be constructed by repeatedly blowing up a Hirzebruch surface at T-fixed points.^[5]

See also

• Projective bundle

References

1. Manetti . Marco . 2005-07-14. Lectures on deformations of complex manifolds . math/0507286.
2. Web site: Algebraic Geometry. Gathmann. Andreas . Fachbereich Mathematik - TU Kaiserslautern .
3. Web site: Section 27.20 (02NB): Twisting by invertible sheaves and relative Proj—The Stacks project. stacks.math.columbia.edu. 2020-05-23.
4. Book: Cox, David A. . Toric varieties . Little . John B. . Schenck . Henry K. . 2011 . American mathematical society . 978-0-8218-4819-7 . Graduate studies in mathematics . Providence (R.I.) . 112.
5. Book: Cox, David A. . Toric varieties . Little . John B. . Schenck . Henry K. . 2011 . American mathematical society . 978-0-8218-4819-7 . Graduate studies in mathematics . Providence (R.I.) . 496.

External links

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