Hirzebruch surface

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

Definition

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

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

where the action of ${\displaystyle \mathbb {C} ^{*}\times \mathbb {C} ^{*}}$ is given byThis 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]: 24where the action of ${\displaystyle \mathbb {C} ^{*}}$ is given byThen, the projectivization ${\displaystyle \mathbb {P} ({\mathcal {O}}\oplus {\mathcal {O}}(-n))}$ is given by another ${\displaystyle \mathbb {C} ^{*}}$-action^[1]: 22 sending an equivalence class ${\displaystyle [l_{0},l_{1},t_{0},t_{1}]\in {\mathcal {O}}\oplus {\mathcal {O}}(-n)}$ toCombining these two actions gives the original quotient up top.

Transition maps

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

Then, the transition maps, induced from the transition maps of ${\displaystyle {\mathcal {O}}\oplus {\mathcal {O}}(-n)}$ give the mapsendingwhere ${\displaystyle X_{i}}$ is the affine coordinate function on ${\displaystyle U_{i}}$.^[2]

Properties

Projective rank 2 bundles over P¹

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

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

Isomorphisms of Hirzebruch surfaces

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

Analysis of associated symmetric algebra

Recall that projective bundles can be constructed using Relative Proj, which is formed from the graded sheaf of algebras

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 tableFor ${\displaystyle i>2}$ the symmetric sheaves are given by

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

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., 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