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

Definition

The Hirzebruch surface ${\displaystyle \Sigma _{n))$ is the ${\displaystyle \mathbb {P} ^{1))$-bundle, called a Projective bundle, over ${\displaystyle \mathbb {P} ^{1))$ associated to the sheaf

$${\displaystyle {\mathcal {O))\oplus {\mathcal {O))(-n).}$$

${\displaystyle {\mathcal {O))(n)}$

${\displaystyle {\mathcal {O))(1)}$

${\displaystyle \Sigma _{0))$

${\displaystyle \Sigma _{1))$

GIT quotient

One method for constructing the Hirzebruch surface is by using a GIT quotient^[1]: 21

$${\displaystyle \Sigma _{n}=(\mathbb {C} ^{2}-\{0\})\times (\mathbb {C} ^{2}-\{0\})/(\mathbb {C} ^{*}\times \mathbb {C} ^{*})}$$

${\displaystyle \mathbb {C} ^{*}\times \mathbb {C} ^{*))$

$${\displaystyle (\lambda ,\mu )\cdot (l_{0},l_{1},t_{0},t_{1})=(\lambda l_{0},\lambda l_{1},\mu t_{0},\lambda ^{-n}\mu t_{1})}$$

${\displaystyle \lambda }$

${\displaystyle \mathbb {C} ^{*))$

${\displaystyle \mathbb {C} ^{2}-\{0\))$

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

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

${\displaystyle {\mathcal {O))\oplus {\mathcal {O))(-n)}$

$${\displaystyle {\mathcal {O))\oplus {\mathcal {O))(-n)=(\mathbb {C} ^{2}-\{0\})\times \mathbb {C} ^{2}/\mathbb {C} ^{*))$$

${\displaystyle \mathbb {C} ^{*))$

$${\displaystyle \lambda \cdot (l_{0},l_{1},t_{0},t_{1})=(\lambda l_{0},\lambda l_{1},\lambda ^{a}t_{0},\lambda ^{0}t_{1}=t_{1})}$$

${\displaystyle \mathbb {P} ({\mathcal {O))\oplus {\mathcal {O))(-n))}$

${\displaystyle \mathbb {C} ^{*))$

${\displaystyle [l_{0},l_{1},t_{0},t_{1}]\in {\mathcal {O))\oplus {\mathcal {O))(-n)}$

$${\displaystyle \mu \cdot [l_{0},l_{1},t_{0},t_{1}]=[l_{0},l_{1},\mu t_{0},\mu t_{1}]}$$

Transition maps

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

$${\displaystyle U_{i}\times \mathbb {P} ^{1))$$

${\displaystyle {\mathcal {O))\oplus {\mathcal {O))(-n)}$

$${\displaystyle U_{0}\times \mathbb {P} ^{1}|_{U_{1))\to U_{1}\times \mathbb {P} ^{1}|_{U_{0))}$$

$${\displaystyle (X_{0},[y_{0}:y_{1}])\mapsto (X_{1},[y_{0}:x_{0}^{n}y_{1}])}$$

${\displaystyle X_{i))$

${\displaystyle U_{i))$

Properties

Projective rank 2 bundles over P¹

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

$${\displaystyle E\cong {\mathcal {O))(a)\oplus {\mathcal {O))(b).}$$

${\displaystyle E={\mathcal {O))(a)\oplus {\mathcal {O))(b)}$

${\displaystyle \Sigma _{b-a))$

${\displaystyle {\mathcal {O))(-a)}$

Isomorphisms of Hirzebruch surfaces

In particular, the above observation gives an isomorphism between ${\displaystyle \Sigma _{n))$ and ${\displaystyle \Sigma _{-n))$ since there is the isomorphism vector bundles

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

Analysis of associated symmetric algebra

Recall that projective bundles can be constructed using Relative Proj, which is formed from the graded sheaf of algebras

$${\displaystyle \bigoplus _{i=0}^{\infty }\operatorname {Sym} ^{i}({\mathcal {O))\oplus {\mathcal {O))(-n))}$$

${\displaystyle \operatorname {Alt} ^{2))$

${\displaystyle {\mathcal {O))\otimes {\mathcal {O))(-n)}$

$${\displaystyle {\begin{aligned}\operatorname {Sym} ^{0}({\mathcal {O))\oplus {\mathcal {O))(-n))&={\mathcal {O))\\\operatorname {Sym} ^{1}({\mathcal {O))\oplus {\mathcal {O))(-n))&={\mathcal {O))\oplus {\mathcal {O))(-n)\\\operatorname {Sym} ^{2}({\mathcal {O))\oplus {\mathcal {O))(-n))&={\mathcal {O))\oplus {\mathcal {O))(-2n)\end{aligned))}$$

${\displaystyle i>2}$

$${\displaystyle {\begin{aligned}\operatorname {Sym} ^{k}({\mathcal {O))\oplus {\mathcal {O))(-n))&=\bigoplus _{i=0}^{k}{\mathcal {O))^{\otimes (n-i)}\otimes {\mathcal {O))(-in)\\&\cong {\mathcal {O))\oplus {\mathcal {O))(-n)\oplus \cdots \oplus {\mathcal {O))(-kn)\end{aligned))}$$

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 P¹). The Picard group is generated by the curve C and one of the fibers, and these generators have intersection matrix

$${\displaystyle {\begin{bmatrix}0&1\\1&-n\end{bmatrix)),}$$