Hirzebruch surface

In mathematics, a Hirzebruch surface is a ruled surface over the projective line. They were studied by Friedrich Hirzebruch (1951).

Definition

The Hirzebruch surface ${\mathrm{\Sigma }}_n$ is the ${\mathbb{\mathbb{P}}}^1$-bundle (a projective bundle) over the projective line ${\mathbb{\mathbb{P}}}^1$, associated to the sheaf$${𝒪}\oplus {𝒪}(-n).$$The notation here means: ${𝒪}(n)$ is the n-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 ${\mathrm{\Sigma }}_0$ is isomorphic to ${\mathbb{\mathbb{P}}}^1\times {\mathbb{\mathbb{P}}}^1$; and ${\mathrm{\Sigma }}_1$ is isomorphic to the projective plane ${\mathbb{\mathbb{P}}}^2$ 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$${\mathrm{\Sigma }}_n=({\mathbb{ℂ}}^2-\{0\})\times ({\mathbb{ℂ}}^2-\{0\})/({\mathbb{ℂ}}^{*}\times {\mathbb{ℂ}}^{*})$$where the action of ${\mathbb{ℂ}}^{*}\times {\mathbb{ℂ}}^{*}$ 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 ${\mathbb{ℂ}}^{*}$ on ${\mathbb{ℂ}}^2-\{0\}$ defining ${\mathbb{\mathbb{P}}}^1$, and the second action is a combination of the construction of a direct sum of line bundles on ${\mathbb{\mathbb{P}}}^1$ and their projectivization. For the direct sum ${𝒪}\oplus {𝒪}(-n)$ this can be given by the quotient variety$${𝒪}\oplus {𝒪}(-n)=({\mathbb{ℂ}}^2-\{0\})\times {\mathbb{ℂ}}^2/{\mathbb{ℂ}}^{*}$$where the action of ${\mathbb{ℂ}}^{*}$ is given by$$\lambda \cdot (l_0,l_1,t_0,t_1)=(\lambda l_0,\lambda l_1,{\lambda }^a{t}_0,{\lambda }^0{t}_1=t_1)$$Then, the projectivization $\mathbb{\mathbb{P}}({𝒪}\oplus {𝒪}(-n))$ is given by another ${\mathbb{ℂ}}^{*}$-action 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 ${\mathbb{\mathbb{P}}}^1$-bundle is by using transition functions. Since affine vector bundles are necessarily trivial, over the charts $U_0,U_1$ of ${\mathbb{\mathbb{P}}}^1$ defined by $x_i\ne 0$ there is the local model of the bundle$$U_i\times {\mathbb{\mathbb{P}}}^1$$Then, the transition maps, induced from the transition maps of ${𝒪}\oplus {𝒪}(-n)$ give the map$$U_0\times {\mathbb{\mathbb{P}}}^1{|}_{U_1}\to U_1\times {\mathbb{\mathbb{P}}}^1{|}_{U_0}$$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 ${\mathbb{\mathbb{P}}}^1$ there are numbers $a,b\in \mathbb{\mathbb{Z}}$ 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 ${\mathrm{\Sigma }}_{b-a}$ since this bundle can be tensored by ${𝒪}(-a)$.

Isomorphisms of Hirzebruch surfaces

In particular, the above observation gives an isomorphism between ${\mathrm{\Sigma }}_n$ and ${\mathrm{\Sigma }}_{-n}$ 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$$\begin{array}{rl}{\mathrm{Sym}}^0({𝒪}\oplus {𝒪}(-n))& ={𝒪}\\ {\mathrm{Sym}}^1({𝒪}\oplus {𝒪}(-n))& ={𝒪}\oplus {𝒪}(-n)\\ {\mathrm{Sym}}^2({𝒪}\oplus {𝒪}(-n))& ={𝒪}\oplus {𝒪}(-2n)\end{array}$$For $i>2$ the symmetric sheaves are given by$$\begin{array}{rl}{\mathrm{Sym}}^k({𝒪}\oplus {𝒪}(-n))& =\underset{i=0}{\overset{k}{⨁}}{{𝒪}}^{\otimes (n-i)}\otimes {𝒪}(-in)\\ & \cong {𝒪}\oplus {𝒪}(-n)\oplus \cdots \oplus {𝒪}(-kn)\end{array}$$

Intersection theory

Hirzebruch surfaces for n > 0 have a special rational curve C on them: The surface is the projective bundle of ${𝒪}(-n)$ and the curve C is the zero section. 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 ${\mathbb{\mathbb{P}}}^1$). The Picard group is generated by the curve C 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 n is even or odd. The Hirzebruch surface Σ_n (n > 1) blown up at a point on the special curve C is isomorphic to Σ_n+1 blown up at a point not on the special curve.

Toric variety

The Hirzebruch surface

${\mathrm{\Sigma }}_n$

can be given an action of the complex torus

$T={\mathbb{ℂ}}^{*}\times {\mathbb{ℂ}}^{*}$

, with one

${\mathbb{ℂ}}^{*}$

acting on the base

${\mathbb{\mathbb{P}}}^1$

with two fixed axis points, and the other

${\mathbb{ℂ}}^{*}$

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

${\mathrm{\Sigma }}_n$

a toric variety. Its associated fan partitions the standard lattice

${\mathbb{\mathbb{Z}}}^2$

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

See also

• Projective bundle

References

• Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225
• Beauville, Arnaud (1996), Complex algebraic surfaces, London Mathematical Society Student Texts, vol. 34 (2nd ed.), Cambridge University Press, ISBN 978-0-521-49510-3, MR1406314
• Hirzebruch, Friedrich (1951), "Über eine Klasse von einfachzusammenhängenden komplexen Mannigfaltigkeiten", Mathematische Annalen, 124: 77–86, doi:10.1007/BF01343552, hdl:21.11116/0000-0004-3A56-B, ISSN 0025-5831, MR 0045384, S2CID 122844063

External links

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