Hirzebruch surface

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: \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^0 t_0=t_0, \lambda^{-n} 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 bundleU_i\times \mathbb{P}^1Then, the transition maps, induced from the transition maps of \mathcal{O}\oplus \mathcal{O}(-n) give the mapU_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 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 thatE \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 \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 Σn (n 1) blown up at a point on the special curve C is isomorphic to Σn+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 \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.

References

References

1. Manetti, Marco. (2005-07-14). "Lectures on deformations of complex manifolds".
2. Gathmann, Andreas. "Algebraic Geometry".
3. "Section 27.20 (02NB): Twisting by invertible sheaves and relative Proj—The Stacks project".
4. (2011). "Toric varieties". American mathematical society.
5. (2011). "Toric varieties". American mathematical society.