Hirzebruch surface

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

The Hirzebruch surface ${\displaystyle \Sigma _{n}}$ is the ${\displaystyle \mathbb {P} ^{1}}$-bundle (a projective bundle) over the projective line ${\displaystyle \mathbb {P} ^{1}}$, associated to the sheaf$${\displaystyle {\mathcal {O}}\oplus {\mathcal {O}}(-n).}$$The notation here means: ${\displaystyle {\mathcal {O}}(n)}$ is the n-th tensor power of the Serre twist sheaf ${\displaystyle {\mathcal {O}}(1)}$, the invertible sheaf or line bundle with associated Cartier divisor a single point. The surface ${\displaystyle \Sigma _{0}}$ is isomorphic to ${\displaystyle \mathbb {P} ^{1}\times \mathbb {P} ^{1}}$; and ${\displaystyle \Sigma _{1}}$ is isomorphic to the projective plane ${\displaystyle \mathbb {P} ^{2}}$ blown up at a point, so it is not minimal.

One method for constructing the Hirzebruch surface is by using a GIT quotient^[1]: 21$${\displaystyle \Sigma _{n}=(\mathbb {C} ^{2}-\{0\})\times (\mathbb {C} ^{2}-\{0\})/(\mathbb {C} ^{*}\times \mathbb {C} ^{*})}$$where the action of ${\displaystyle \mathbb {C} ^{*}\times \mathbb {C} ^{*}}$ is given by$${\displaystyle (\lambda ,\mu )\cdot (l_{0},l_{1},t_{0},t_{1})=(\lambda l_{0},\lambda l_{1},\mu t_{0},\lambda ^{-n}\mu t_{1})}$$This action can be interpreted as the action of ${\displaystyle \lambda }$ on the first two factors comes from the action of ${\displaystyle \mathbb {C} ^{*}}$ on ${\displaystyle \mathbb {C} ^{2}-\{0\}}$ defining ${\displaystyle \mathbb {P} ^{1}}$, and the second action is a combination of the construction of a direct sum of line bundles on ${\displaystyle \mathbb {P} ^{1}}$ and their projectivization. For the direct sum ${\displaystyle {\mathcal {O}}\oplus {\mathcal {O}}(-n)}$ this can be given by the quotient variety^[1]: 24$${\displaystyle {\mathcal {O}}\oplus {\mathcal {O}}(-n)=(\mathbb {C} ^{2}-\{0\})\times \mathbb {C} ^{2}/\mathbb {C} ^{*}}$$where the action of ${\displaystyle \mathbb {C} ^{*}}$ is given by$${\displaystyle \lambda \cdot (l_{0},l_{1},t_{0},t_{1})=(\lambda l_{0},\lambda l_{1},\lambda ^{a}t_{0},\lambda ^{0}t_{1}=t_{1})}$$Then, the projectivization ${\displaystyle \mathbb {P} ({\mathcal {O}}\oplus {\mathcal {O}}(-n))}$ is given by another ${\displaystyle \mathbb {C} ^{*}}$-action^[1]: 22 sending an equivalence class ${\displaystyle [l_{0},l_{1},t_{0},t_{1}]\in {\mathcal {O}}\oplus {\mathcal {O}}(-n)}$ to$${\displaystyle \mu \cdot [l_{0},l_{1},t_{0},t_{1}]=[l_{0},l_{1},\mu t_{0},\mu t_{1}]}$$Combining these two actions gives the original quotient up top.

One way to construct this ${\displaystyle \mathbb {P} ^{1}}$-bundle is by using transition functions. Since affine vector bundles are necessarily trivial, over the charts ${\displaystyle U_{0},U_{1}}$ of ${\displaystyle \mathbb {P} ^{1}}$ defined by ${\displaystyle x_{i}\neq 0}$ there is the local model of the bundle$${\displaystyle U_{i}\times \mathbb {P} ^{1}}$$Then, the transition maps, induced from the transition maps of ${\displaystyle {\mathcal {O}}\oplus {\mathcal {O}}(-n)}$ give the map$${\displaystyle U_{0}\times \mathbb {P} ^{1}|_{U_{1}}\to U_{1}\times \mathbb {P} ^{1}|_{U_{0}}}$$sending$${\displaystyle (X_{0},[y_{0}:y_{1}])\mapsto (X_{1},[y_{0}:x_{0}^{n}y_{1}])}$$where ${\displaystyle X_{i}}$ is the affine coordinate function on ${\displaystyle U_{i}}$.^[2]

Note that by Grothendieck's theorem, for any rank 2 vector bundle ${\displaystyle E}$ on ${\displaystyle \mathbb {P} ^{1}}$ there are numbers ${\displaystyle a,b\in \mathbb {Z} }$ such that$${\displaystyle E\cong {\mathcal {O}}(a)\oplus {\mathcal {O}}(b).}$$As taking the projective bundle is invariant under tensoring by a line bundle,^[3] the ruled surface associated to ${\displaystyle E={\mathcal {O}}(a)\oplus {\mathcal {O}}(b)}$ is the Hirzebruch surface ${\displaystyle \Sigma _{b-a}}$ since this bundle can be tensored by ${\displaystyle {\mathcal {O}}(-a)}$.

In particular, the above observation gives an isomorphism between ${\displaystyle \Sigma _{n}}$ and ${\displaystyle \Sigma _{-n}}$ since there is the isomorphism vector bundles$${\displaystyle {\mathcal {O}}(n)\otimes ({\mathcal {O}}\oplus {\mathcal {O}}(-n))\cong {\mathcal {O}}(n)\oplus {\mathcal {O}}}$$

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

Hirzebruch surfaces for n > 0 have a special rational curve C on them: The surface is the projective bundle of ${\displaystyle {\mathcal {O}}(-n)}$ 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 ${\displaystyle \mathbb {P} ^{1}}$). The Picard group is generated by the curve C and one of the fibers, and these generators have intersection matrix$${\displaystyle {\begin{bmatrix}0&1\\1&-n\end{bmatrix}},}$$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.

The Hirzebruch surface ${\displaystyle \Sigma _{n}}$ can be given an action of the complex torus ${\displaystyle T=\mathbb {C} ^{*}\times \mathbb {C} ^{*}}$, with one ${\displaystyle \mathbb {C} ^{*}}$ acting on the base ${\displaystyle \mathbb {P} ^{1}}$ with two fixed axis points, and the other ${\displaystyle \mathbb {C} ^{*}}$ 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 ${\displaystyle \Sigma _{n}}$ a toric variety. Its associated fan partitions the standard lattice ${\displaystyle \mathbb {Z} ^{2}}$ into four cones (each corresponding to a coordinate chart), separated by the rays along the four vectors:^[4]

${\displaystyle (1,0),(0,1),(0,-1),(-1,n).}$

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

• Projective bundle

• 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

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