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 $\Sigma _{n}$ 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^[1]^: 21: $\Sigma _{n}=(\mathbb {C} ^{2}-\{0\})\times (\mathbb {C} ^{2}-\{0\})/(\mathbb {C} ^{*}\times \mathbb {C} ^{*})$ where the action of $\mathbb {C} ^{*}\times \mathbb {C} ^{*}$ 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 $\mathbb {C} ^{*}$ on $\mathbb {C} ^{2}-\{0\}$ 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^[1]^: 24 ${\mathcal {O}}\oplus {\mathcal {O}}(-n)=(\mathbb {C} ^{2}-\{0\})\times \mathbb {C} ^{2}/\mathbb {C} ^{*}$ where the action of $\mathbb {C} ^{*}$ 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 $\mathbb {C} ^{*}$ -action^[1]^: 22 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 $U_{i}$ .^[2]

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 that $E\cong {\mathcal {O}}(a)\oplus {\mathcal {O}}(b).$ As taking the projective bundle is invariant under tensoring by a line bundle,^[3] the ruled surface associated to $E={\mathcal {O}}(a)\oplus {\mathcal {O}}(b)$ 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 ${\begin{aligned}\operatorname {Sym} ^{0}({\mathcal {O}}\oplus {\mathcal {O}}(-n))&={\mathcal {O}}\\\operatorname {Sym} ^{1}({\mathcal {O}}\oplus {\mathcal {O}}(-n))&={\mathcal {O}}\oplus {\mathcal {O}}(-n)\\\operatorname {Sym} ^{2}({\mathcal {O}}\oplus {\mathcal {O}}(-n))&={\mathcal {O}}\oplus {\mathcal {O}}(-2n)\end{aligned}}$ For $i>2$ the symmetric sheaves are given by ${\begin{aligned}\operatorname {Sym} ^{k}({\mathcal {O}}\oplus {\mathcal {O}}(-n))&=\bigoplus _{i=0}^{k}{\mathcal {O}}^{\otimes (n-i)}\otimes {\mathcal {O}}(-in)\\&\cong {\mathcal {O}}\oplus {\mathcal {O}}(-n)\oplus \cdots \oplus {\mathcal {O}}(-kn)\end{aligned}}$

Intersection theory

Hirzebruch surfaces for $n > 0$ have a special rational curve $C$ on them: The surface is the projective bundle of ${\mathcal {O}}(-n)$ 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 $Σ n$ ( $n > 1$ ) 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.

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 ${\textstyle {\mathcal {O}}\oplus {\mathcal {O}}(-n)}$ , 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:^[4]

$(1,0),(0,1),(0,-1),(-1,n).$

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.^[5]

References

^ ^a ^b ^c Manetti, Marco (2005-07-14). "Lectures on deformations of complex manifolds". arXiv:math/0507286.
^ Gathmann, Andreas. "Algebraic Geometry" (PDF). Fachbereich Mathematik - TU Kaiserslautern.
^ "Section 27.20 (02NB): Twisting by invertible sheaves and relative Proj—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-05-23.
^ Cox, David A.; Little, John B.; Schenck, Henry K. (2011). Toric varieties. Graduate studies in mathematics. Providence (R.I.): American mathematical society. p. 112. ISBN 978-0-8218-4819-7.
^ Cox, David A.; Little, John B.; Schenck, Henry K. (2011). Toric varieties. Graduate studies in mathematics. Providence (R.I.): American mathematical society. p. 496. ISBN 978-0-8218-4819-7.

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

[:0-1] 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.

[4] Cox, David A.; Little, John B.; Schenck, Henry K. (2011). Toric varieties. Graduate studies in mathematics. Providence (R.I.): American mathematical society. p. 112. ISBN 978-0-8218-4819-7.

[5] Cox, David A.; Little, John B.; Schenck, Henry K. (2011). Toric varieties. Graduate studies in mathematics. Providence (R.I.): American mathematical society. p. 496. ISBN 978-0-8218-4819-7.

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