Hirzebruch surface

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

Definition
The Hirzebruch surface $$\Sigma_n$$ is the $$\mathbb{P}^1$$-bundle, called a Projective bundle, over $$\mathbb{P}^1$$ 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 $P^{1} × P^{1}$, and $$\Sigma_1$$ is isomorphic to $P^{2}$ blown up at a point so is not minimal.

GIT quotient
One method for constructing the Hirzebruch surface is by using a GIT quotient $$\Sigma_n = (\Complex^2-\{0\})\times (\Complex^2-\{0\})/(\Complex^*\times\Complex^*)$$where the action of $$\Complex^*\times\Complex^*$$ 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 $$\Complex^*$$ on $$\Complex^2 - \{0\}$$ 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 $$\mathcal{O}\oplus \mathcal{O}(-n) = (\Complex^2-\{0\})\times \Complex^2/\Complex^*$$where the action of $$\Complex^*$$ 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 $$\Complex^*$$-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 $$U_i$$.

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$$\begin{align} \operatorname{Sym}^0(\mathcal{O}\oplus \mathcal{O}(-n)) &= \mathcal{O} \\ \operatorname{Sym}^1(\mathcal{O}\oplus \mathcal{O}(-n)) &= \mathcal{O} \oplus \mathcal{O}(-n) \\ \operatorname{Sym}^2(\mathcal{O}\oplus \mathcal{O}(-n)) &= \mathcal{O} \oplus \mathcal{O}(-2n) \end{align}$$For $$i > 2$$ the symmetric sheaves are given by$$\begin{align} \operatorname{Sym}^k(\mathcal{O}\oplus \mathcal{O}(-n)) &= \bigoplus_{i=0}^k \mathcal{O}^{\otimes (n-i)}\otimes \mathcal{O}(-in) \\ &\cong \mathcal{O}\oplus \mathcal{O}(-n) \oplus \cdots \oplus \mathcal{O}(-kn) \end{align}$$

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 $P^{1}$). 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.