Hirzebruch surface

Template:Short description In mathematics, a Hirzebruch surface is a ruled surface over the projective line. They were studied by Template:Harvs.

Definition

The Hirzebruch surface ${\Sigma }_n$ is the ${\mathbb{P}}^1$-bundle (a projective bundle) over the projective line ${\mathbb{P}}^1$, associated to the sheaf$$𝒪\oplus 𝒪(-n).$$The notation here means: $𝒪(n)$ is the Template:Mvar-th tensor power of the Serre twist sheaf $𝒪(1)$, the invertible sheaf or line bundle with associated Cartier divisor 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:Template:Rp $${\Sigma }_n=({ℂ}^2-\{0\})\times ({ℂ}^2-\{0\})/({ℂ}^{\ast }\times {ℂ}^{\ast })$$ where the action of ${ℂ}^{\ast }\times {ℂ}^{\ast }$ is given by $$(\lambda ,\mu )\cdot (l_0,l_1,t_0,t_1)=(\lambda l_0,\lambda l_1,\mu t_0,{\lambda }^{-n}\mu t_1).$$ This action can be interpreted as the action of $\lambda$ on the first two factors comes from the action of ${ℂ}^{\ast }$ on ${ℂ}^2-\{0\}$ defining ${\mathbb{P}}^1$, and the second action is a combination of the construction of a direct sum of line bundles on ${\mathbb{P}}^1$ and their projectivization. For the direct sum $𝒪\oplus 𝒪(-n)$ this can be given by the quotient varietyTemplate:Rp$$𝒪\oplus 𝒪(-n)=({ℂ}^2-\{0\})\times {ℂ}^2/{ℂ}^{\ast }$$where the action of ${ℂ}^{\ast }$ is given by$$\lambda \cdot (l_0,l_1,t_0,t_1)=(\lambda l_0,\lambda l_1,{\lambda }^0{t}_0=t_0,{\lambda }^{-n}{t}_1)$$Then, the projectivization $\mathbb{P}(𝒪\oplus 𝒪(-n))$ is given by another ${ℂ}^{\ast }$-actionTemplate:Rp sending an equivalence class $[l_0,l_1,t_0,t_1]\in 𝒪\oplus 𝒪(-n)$ to$$\mu \cdot [l_0,l_1,t_0,t_1]=[l_0,l_1,\mu t_0,\mu t_1]$$Combining these two actions gives the original quotient up top.

Transition maps

One way to construct this ${\mathbb{P}}^1$-bundle is by using transition functions. Since affine vector bundles are necessarily trivial, over the charts $U_0,U_1$ of ${\mathbb{P}}^1$ defined by $x_i\ne 0$ there is the local model of the bundle$$U_i\times {\mathbb{P}}^1$$Then, the transition maps, induced from the transition maps of $𝒪\oplus 𝒪(-n)$ give the map$$U_0\times {\mathbb{P}}^1{|}_{U_1}\to U_1\times {\mathbb{P}}^1{|}_{U_0}$$sending$$(X_0,[y_0:y_1])\mapsto (X_1,[y_0:x_0^n{y}_1])$$where $X_i$ 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 that$$E\cong 𝒪(a)\oplus 𝒪(b).$$As taking the projective bundle is invariant under tensoring by a line bundle, the ruled surface associated to $E=𝒪(a)\oplus 𝒪(b)$ is the Hirzebruch surface ${\Sigma }_{b-a}$ since this bundle can be tensored by $𝒪(-a)$.

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$$𝒪(n)\otimes (𝒪\oplus 𝒪(-n))\cong 𝒪(n)\oplus 𝒪$$

Analysis of associated symmetric algebra

Recall that projective bundles can be constructed using Relative Proj, which is formed from the graded sheaf of algebras$$\underset{i=0}{\overset{\mathrm{\infty }}{⨁}}{\mathrm{Sym}}^i(𝒪\oplus 𝒪(-n))$$The first few symmetric modules are special since there is a non-trivial anti-symmetric ${\mathrm{Alt}}^2$-module $𝒪\otimes 𝒪(-n)$. These sheaves are summarized in the table$$\begin{array}{cc}{\mathrm{Sym}}^0(𝒪\oplus 𝒪(-n))& =𝒪\\ {\mathrm{Sym}}^1(𝒪\oplus 𝒪(-n))& =𝒪\oplus 𝒪(-n)\\ {\mathrm{Sym}}^2(𝒪\oplus 𝒪(-n))& =𝒪\oplus 𝒪(-2n)\end{array}$$For $i>2$ the symmetric sheaves are given by$$\begin{array}{cc}{\mathrm{Sym}}^k(𝒪\oplus 𝒪(-n))& =\underset{i=0}{\overset{k}{⨁}}{𝒪}^{\otimes (n-i)}\otimes 𝒪(-in)\\ & \cong 𝒪\oplus 𝒪(-n)\oplus \cdots \oplus 𝒪(-kn)\end{array}$$

Intersection theory

Hirzebruch surfaces for Template:Math have a special rational curve Template:Math on them: The surface is the projective bundle of $𝒪(-n)$ and the curve Template:Math is the zero section. This curve has self-intersection number Template:Math, 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 is generated by the curve Template:Math and one of the fibers, and these generators have intersection matrix$$\left[\begin{array}{cc}0& 1\\ 1& -n\end{array}\right],$$so the bilinear form is two dimensional unimodular, and is even or odd depending on whether Template:Mvar is even or odd. The Hirzebruch surface Template:Math (Template:Math) blown up at a point on the special curve Template:Math is isomorphic to Template:Math 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

$T={ℂ}^{\ast }\times {ℂ}^{\ast }$

, with one

${ℂ}^{\ast }$

acting on the base

${\mathbb{P}}^1$

with two fixed axis points, and the other

${ℂ}^{\ast }$

acting on the fibers of the vector bundle

$𝒪\oplus 𝒪(-n)$

, 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

${\mathbb{Z}}^2$

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 ${\mathbb{P}}^2$ can be constructed by repeatedly blowing up a Hirzebruch surface at T-fixed points.

See also

• Projective bundle

References

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External links

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