Hirzebruch surface

Short description: Ruled surface over the projective line

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

Definition

The Hirzebruch surface [math]\displaystyle{ \Sigma_n }[/math] is the [math]\displaystyle{ \mathbb{P}^1 }[/math]-bundle, called a Projective bundle, over [math]\displaystyle{ \mathbb{P}^1 }[/math] associated to the sheaf[math]\displaystyle{ \mathcal{O}\oplus \mathcal{O}(-n). }[/math]The notation here means: [math]\displaystyle{ \mathcal{O}(n) }[/math] is the n-th tensor power of the Serre twist sheaf [math]\displaystyle{ \mathcal{O}(1) }[/math], the invertible sheaf or line bundle with associated Cartier divisor a single point. The surface [math]\displaystyle{ \Sigma_0 }[/math] is isomorphic to P¹ × P¹, and [math]\displaystyle{ \Sigma_1 }[/math] is isomorphic to P² blown up at a point so is not minimal.

GIT quotient

One method for constructing the Hirzebruch surface is by using a GIT quotient[math]\displaystyle{ \Sigma_n = (\Complex^2-\{0\})\times (\Complex^2-\{0\})/(\Complex^*\times\Complex^*) }[/math]where the action of [math]\displaystyle{ \Complex^*\times\Complex^* }[/math] is given by[math]\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) }[/math]This action can be interpreted as the action of [math]\displaystyle{ \lambda }[/math] on the first two factors comes from the action of [math]\displaystyle{ \Complex^* }[/math] on [math]\displaystyle{ \Complex^2 - \{0\} }[/math] defining [math]\displaystyle{ \mathbb{P}^1 }[/math], and the second action is a combination of the construction of a direct sum of line bundles on [math]\displaystyle{ \mathbb{P}^1 }[/math] and their projectivization. For the direct sum [math]\displaystyle{ \mathcal{O}\oplus \mathcal{O}(-n) }[/math] this can be given by the quotient variety[math]\displaystyle{ \mathcal{O}\oplus \mathcal{O}(-n) = (\Complex^2-\{0\})\times \Complex^2/\Complex^* }[/math]where the action of [math]\displaystyle{ \Complex^* }[/math] is given by[math]\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) }[/math]Then, the projectivization [math]\displaystyle{ \mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-n)) }[/math] is given by another [math]\displaystyle{ \Complex^* }[/math]-action sending an equivalence class [math]\displaystyle{ [l_0,l_1,t_0,t_1] \in\mathcal{O}\oplus\mathcal{O}(-n) }[/math] to[math]\displaystyle{ \mu \cdot [l_0,l_1,t_0,t_1] = [l_0,l_1,\mu t_0,\mu t_1] }[/math]Combining these two actions gives the original quotient up top.

Transition maps

One way to construct this [math]\displaystyle{ \mathbb{P}^1 }[/math]-bundle is by using transition functions. Since affine vector bundles are necessarily trivial, over the charts [math]\displaystyle{ U_0,U_1 }[/math] of [math]\displaystyle{ \mathbb{P}^1 }[/math] defined by [math]\displaystyle{ x_i \neq 0 }[/math] there is the local model of the bundle[math]\displaystyle{ U_i\times \mathbb{P}^1 }[/math]Then, the transition maps, induced from the transition maps of [math]\displaystyle{ \mathcal{O}\oplus \mathcal{O}(-n) }[/math] give the map[math]\displaystyle{ U_0\times\mathbb{P}^1|_{U_1} \to U_1\times\mathbb{P}^1|_{U_0} }[/math]sending[math]\displaystyle{ (X_0, [y_0:y_1]) \mapsto (X_1, [y_0:x_0^n y_1]) }[/math]where [math]\displaystyle{ X_i }[/math] is the affine coordinate function on [math]\displaystyle{ U_i }[/math].

Properties

Projective rank 2 bundles over P¹

Note that by Grothendieck's theorem, for any rank 2 vector bundle [math]\displaystyle{ E }[/math] on [math]\displaystyle{ \mathbb P^1 }[/math] there are numbers [math]\displaystyle{ a,b \in \mathbb Z }[/math] such that[math]\displaystyle{ E \cong \mathcal{O}(a)\oplus \mathcal{O}(b). }[/math]As taking the projective bundle is invariant under tensoring by a line bundle, the ruled surface associated to [math]\displaystyle{ E = \mathcal O(a) \oplus \mathcal O(b) }[/math] is the Hirzebruch surface [math]\displaystyle{ \Sigma_{b-a} }[/math] since this bundle can be tensored by [math]\displaystyle{ \mathcal{O}(-a) }[/math].

Isomorphisms of Hirzebruch surfaces

In particular, the above observation gives an isomorphism between [math]\displaystyle{ \Sigma_n }[/math] and [math]\displaystyle{ \Sigma_{-n} }[/math] since there is the isomorphism vector bundles[math]\displaystyle{ \mathcal{O}(n)\otimes(\mathcal{O} \oplus \mathcal{O}(-n)) \cong \mathcal{O}(n) \oplus \mathcal{O} }[/math]

Analysis of associated symmetric algebra

Recall that projective bundles can be constructed using Relative Proj, which is formed from the graded sheaf of algebras[math]\displaystyle{ \bigoplus_{i=0}^\infty \operatorname{Sym}^i(\mathcal{O}\oplus \mathcal{O}(-n)) }[/math]The first few symmetric modules are special since there is a non-trivial anti-symmetric [math]\displaystyle{ \operatorname{Alt}^2 }[/math]-module [math]\displaystyle{ \mathcal{O}\otimes \mathcal{O}(-n) }[/math]. These sheaves are summarized in the table[math]\displaystyle{ \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} }[/math]For [math]\displaystyle{ i \gt 2 }[/math] the symmetric sheaves are given by[math]\displaystyle{ \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} }[/math]

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[math]\displaystyle{ \begin{bmatrix}0 & 1 \\ 1 & -n \end{bmatrix} , }[/math]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.

See also

• Projective bundle

References

• Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3 
• Beauville, Arnaud (1996), Complex algebraic surfaces, London Mathematical Society Student Texts, 34 (2nd ed.), Cambridge University Press, MR1406314, ISBN 978-0-521-49510-3 
• Hirzebruch, Friedrich (1951), "Über eine Klasse von einfachzusammenhängenden komplexen Mannigfaltigkeiten", Mathematische Annalen 124: 77–86, doi:10.1007/BF01343552, ISSN 0025-5831 

External links

• Manifold Atlas
• https://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/alggeom-2002-c10.pdf
• https://mathoverflow.net/q/122952