Hirzebruch surface

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

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

where ${\displaystyle X_{i}}$ is the affine coordinate function on ${\displaystyle U_{i}}$.[2]

Note that the projective bundle

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

is equivalent to a Hirzebruch surface since projective bundles are invariant after tensoring by a line bundle.[3] In particular, this is associated to the Hirzebruch surface ${\displaystyle \Sigma _{b-a}}$ since this bundle can be tensored by the line bundle ${\displaystyle {\mathcal {O}}(-a)}$.

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

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