Hirzebruch surface

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 Σ n {\displaystyle \Sigma _{n}} is the P 1 {\displaystyle \mathbb {P} ^{1}} -bundle, called a Projective bundle, over P 1 {\displaystyle \mathbb {P} ^{1}} associated to the sheaf

O ⊕ O ( − n ) . {\displaystyle {\mathcal {O}}\oplus {\mathcal {O}}(-n).}
The notation here means: O ( n ) {\displaystyle {\mathcal {O}}(n)} is the n-th tensor power of the Serre twist sheaf O ( 1 ) {\displaystyle {\mathcal {O}}(1)} , the invertible sheaf or line bundle with associated Cartier divisor a single point. The surface Σ 0 {\displaystyle \Sigma _{0}} is isomorphic to P1 × P1, and Σ 1 {\displaystyle \Sigma _{1}} 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[1]: 21 

Σ n = ( C 2 − { 0 } ) × ( C 2 − { 0 } ) / ( C ∗ × C ∗ ) {\displaystyle \Sigma _{n}=(\mathbb {C} ^{2}-\{0\})\times (\mathbb {C} ^{2}-\{0\})/(\mathbb {C} ^{*}\times \mathbb {C} ^{*})}
where the action of C ∗ × C ∗ {\displaystyle \mathbb {C} ^{*}\times \mathbb {C} ^{*}} is given by
( λ , μ ) ⋅ ( l 0 , l 1 , t 0 , t 1 ) = ( λ l 0 , λ l 1 , μ t 0 , λ − n μ t 1 ) {\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 λ {\displaystyle \lambda } on the first two factors comes from the action of C ∗ {\displaystyle \mathbb {C} ^{*}} on C 2 − { 0 } {\displaystyle \mathbb {C} ^{2}-\{0\}} defining P 1 {\displaystyle \mathbb {P} ^{1}} , and the second action is a combination of the construction of a direct sum of line bundles on P 1 {\displaystyle \mathbb {P} ^{1}} and their projectivization. For the direct sum O ⊕ O ( − n ) {\displaystyle {\mathcal {O}}\oplus {\mathcal {O}}(-n)} this can be given by the quotient variety[1]: 24 
O ⊕ O ( − n ) = ( C 2 − { 0 } ) × C 2 / C ∗ {\displaystyle {\mathcal {O}}\oplus {\mathcal {O}}(-n)=(\mathbb {C} ^{2}-\{0\})\times \mathbb {C} ^{2}/\mathbb {C} ^{*}}
where the action of C ∗ {\displaystyle \mathbb {C} ^{*}} is given by
λ ⋅ ( l 0 , l 1 , t 0 , t 1 ) = ( λ l 0 , λ l 1 , λ a t 0 , λ 0 t 1 = t 1 ) {\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 P ( O ⊕ O ( − n ) ) {\displaystyle \mathbb {P} ({\mathcal {O}}\oplus {\mathcal {O}}(-n))} is given by another C ∗ {\displaystyle \mathbb {C} ^{*}} -action[1]: 22  sending an equivalence class [ l 0 , l 1 , t 0 , t 1 ] ∈ O ⊕ O ( − n ) {\displaystyle [l_{0},l_{1},t_{0},t_{1}]\in {\mathcal {O}}\oplus {\mathcal {O}}(-n)} to
μ ⋅ [ l 0 , l 1 , t 0 , t 1 ] = [ l 0 , l 1 , μ t 0 , μ t 1 ] {\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.

Transition maps

One way to construct this P 1 {\displaystyle \mathbb {P} ^{1}} -bundle is by using transition functions. Since affine vector bundles are necessarily trivial, over the charts U 0 , U 1 {\displaystyle U_{0},U_{1}} of P 1 {\displaystyle \mathbb {P} ^{1}} defined by x i ≠ 0 {\displaystyle x_{i}\neq 0} there is the local model of the bundle

U i × P 1 {\displaystyle U_{i}\times \mathbb {P} ^{1}}
Then, the transition maps, induced from the transition maps of O ⊕ O ( − n ) {\displaystyle {\mathcal {O}}\oplus {\mathcal {O}}(-n)} give the map
U 0 × P 1 | U 1 → U 1 × P 1 | U 0 {\displaystyle U_{0}\times \mathbb {P} ^{1}|_{U_{1}}\to U_{1}\times \mathbb {P} ^{1}|_{U_{0}}}
sending
( X 0 , [ y 0 : y 1 ] ) ↦ ( X 1 , [ y 0 : x 0 n y 1 ] ) {\displaystyle (X_{0},[y_{0}:y_{1}])\mapsto (X_{1},[y_{0}:x_{0}^{n}y_{1}])}
where X i {\displaystyle X_{i}} is the affine coordinate function on U i {\displaystyle U_{i}} .[2]

Properties

Projective rank 2 bundles over P1

Note that by Grothendieck's theorem, for any rank 2 vector bundle E {\displaystyle E} on P 1 {\displaystyle \mathbb {P} ^{1}} there are numbers a , b ∈ Z {\displaystyle a,b\in \mathbb {Z} } such that

E ≅ O ( a ) ⊕ O ( b ) . {\displaystyle E\cong {\mathcal {O}}(a)\oplus {\mathcal {O}}(b).}
As taking the projective bundle is invariant under tensoring by a line bundle,[3] the ruled surface associated to E = O ( a ) ⊕ O ( b ) {\displaystyle E={\mathcal {O}}(a)\oplus {\mathcal {O}}(b)} is the Hirzebruch surface Σ b − a {\displaystyle \Sigma _{b-a}} since this bundle can be tensored by O ( − a ) {\displaystyle {\mathcal {O}}(-a)} .

Isomorphisms of Hirzebruch surfaces

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

O ( n ) ⊗ ( O ⊕ O ( − n ) ) ≅ O ( n ) ⊕ O {\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

⨁ i = 0 ∞ Sym i ⁡ ( O ⊕ O ( − n ) ) {\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 Alt 2 {\displaystyle \operatorname {Alt} ^{2}} -module O ⊗ O ( − n ) {\displaystyle {\mathcal {O}}\otimes {\mathcal {O}}(-n)} . These sheaves are summarized in the table
Sym 0 ⁡ ( O ⊕ O ( − n ) ) = O Sym 1 ⁡ ( O ⊕ O ( − n ) ) = O ⊕ O ( − n ) Sym 2 ⁡ ( O ⊕ O ( − n ) ) = O ⊕ O ( − 2 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}}}
For i > 2 {\displaystyle i>2} the symmetric sheaves are given by
Sym k ⁡ ( O ⊕ O ( − n ) ) = ⨁ i = 0 k O ⊗ ( n − i ) ⊗ O ( − i n ) ≅ O ⊕ O ( − n ) ⊕ ⋯ ⊕ O ( − k n ) {\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

[ 0 1 1 − n ] , {\displaystyle {\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.

See also

  • Projective bundle

References

  1. ^ a b c Manetti, Marco (2005-07-14). "Lectures on deformations of complex manifolds". arXiv:math/0507286.
  2. ^ Gathmann, Andreas. "Algebraic Geometry" (PDF). Fachbereich Mathematik - TU Kaiserslautern.
  3. ^ "Section 27.20 (02NB): Twisting by invertible sheaves and relative Proj—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-05-23.
  • 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., vol. 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225
  • Beauville, Arnaud (1996), Complex algebraic surfaces, London Mathematical Society Student Texts, vol. 34 (2nd ed.), Cambridge University Press, ISBN 978-0-521-49510-3, MR1406314
  • Hirzebruch, Friedrich (1951), "Über eine Klasse von einfachzusammenhängenden komplexen Mannigfaltigkeiten", Mathematische Annalen, 124: 77–86, doi:10.1007/BF01343552, hdl:21.11116/0000-0004-3A56-B, ISSN 0025-5831, MR 0045384, S2CID 122844063

External links

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