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 $\Sigma _{n}$ is the $\mathbb {P} ^{1}$-bundle (a projective bundle) over the projective line $\mathbb {P} ^{1}$, associated to the sheaf$${\mathcal {O}}\oplus {\mathcal {O}}(-n).$$The notation here means: ${\mathcal {O}}(n)$ is the n-th tensor power of the Serre twist sheaf ${\mathcal {O}}(1)$, 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$$\Sigma _{n}=(\mathbb {C} ^{2}-\{0\})\times (\mathbb {C} ^{2}-\{0\})/(\mathbb {C} ^{*}\times \mathbb {C} ^{*})$$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 $\mathbb {P} ^{1}$, 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 ${\mathcal {O}}\oplus {\mathcal {O}}(-n)$ this can be given by the quotient variety$${\mathcal {O}}\oplus {\mathcal {O}}(-n)=(\mathbb {C} ^{2}-\{0\})\times \mathbb {C} ^{2}/\mathbb {C} ^{*}$$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 $\mathbb {P} ({\mathcal {O}}\oplus {\mathcal {O}}(-n))$ is given by another $\mathbb {C} ^{*}$-action sending an equivalence class $[l_{0},l_{1},t_{0},t_{1}]\in {\mathcal {O}}\oplus {\mathcal {O}}(-n)$ 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 $x_{i}\neq 0$ there is the local model of the bundle$$U_{i}\times \mathbb {P} ^{1}$$Then, the transition maps, induced from the transition maps of ${\mathcal {O}}\oplus {\mathcal {O}}(-n)$ 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$$E\cong {\mathcal {O}}(a)\oplus {\mathcal {O}}(b).$$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 $\Sigma _{b-a}$ since this bundle can be tensored by ${\mathcal {O}}(-a)$.

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$${\mathcal {O}}(n)\otimes ({\mathcal {O}}\oplus {\mathcal {O}}(-n))\cong {\mathcal {O}}(n)\oplus {\mathcal {O}}$$

Analysis of associated symmetric algebra

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

Intersection theory

Hirzebruch surfaces for n > 0 have a special rational curve C on them: The surface is the projective bundle of ${\mathcal {O}}(-n)$ 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 $\mathbb {P} ^{1}$). The Picard group is generated by the curve C and one of the fibers, and these generators have intersection matrix$${\begin{bmatrix}0&1\\1&-n\end{bmatrix}},$$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 $\Sigma _{n}$ can be given an action of the complex torus $T=\mathbb {C} ^{*}\times \mathbb {C} ^{*}$, with one $\mathbb {C} ^{*}$ acting on the base $\mathbb {P} ^{1}$ with two fixed axis points, and the other $\mathbb {C} ^{*}$ acting on the fibers of the vector bundle ${\textstyle {\mathcal {O}}\oplus {\mathcal {O}}(-n)}$, 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

• 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