# 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:$^{:21}$ $\Sigma _{n}=(\mathbb {C} ^{2}-\{0\})\times (\mathbb {C} ^{2}-\{0\})/(\mathbb {C} ^{*}\times \mathbb {C} ^{*})$ 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$^{:24}$${\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$\lambda \cdot (l_{0},l_{1},t_{0},t_{1})=(\lambda l_{0},\lambda l_{1},\lambda ^{0}t_{0}=t_{0},\lambda ^{-n}t_{1})$Then, the projectivization $\mathbb {P} ({\mathcal {O}}\oplus {\mathcal {O}}(-n))$ is given by another $\mathbb {C} ^{*}$-action$^{:22}$ 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 P$^1$

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 O⊕O(−n){\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.