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
is the
-bundle, called a Projective bundle, over
associated to the sheaf
![{\displaystyle {\mathcal {O}}\oplus {\mathcal {O}}(-n).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffc5da62b415a393a21ae9c577006fec4c8c7c5e)
The notation here means:
![{\displaystyle {\mathcal {O}}(n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c7bbe0124ae81792773344bc8709fc2f9c9910d)
is the
n-th tensor power of the Serre twist sheaf
![{\displaystyle {\mathcal {O}}(1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5aeb15c854068604d35a2dd82a925899fafd3690)
, the invertible sheaf or line bundle with associated Cartier divisor a single point. The surface
![{\displaystyle \Sigma _{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a69031c973bdef03ce24599b02b1cff9108a8421)
is isomorphic to
P1 × P1, and
![{\displaystyle \Sigma _{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d37ee4ac26b75c4bca37e546fcc9859f3e45adf9)
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
![{\displaystyle \Sigma _{n}=(\mathbb {C} ^{2}-\{0\})\times (\mathbb {C} ^{2}-\{0\})/(\mathbb {C} ^{*}\times \mathbb {C} ^{*})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11f32ac383566df5681eb0b0e5d3c5ab668dbba3)
where the action of
![{\displaystyle \mathbb {C} ^{*}\times \mathbb {C} ^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5db26829cfbad04fd159246bc6f2898342217e4b)
is given by
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42d84a6c021a6943ddfdf0274497d6eb58479245)
This action can be interpreted as the action of
![{\displaystyle \lambda }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a)
on the first two factors comes from the action of
![{\displaystyle \mathbb {C} ^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12e4a3fc8042b1c73d31cfe579c92d34f1255137)
on
![{\displaystyle \mathbb {C} ^{2}-\{0\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9b172236a92afc1708562de4bf14d4ea86fcb51)
defining
![{\displaystyle \mathbb {P} ^{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e38808c13ecc929e128079b70c1642554e2a4b3)
, and the second action is a combination of the construction of a direct sum of line bundles on
![{\displaystyle \mathbb {P} ^{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e38808c13ecc929e128079b70c1642554e2a4b3)
and their projectivization. For the direct sum
![{\displaystyle {\mathcal {O}}\oplus {\mathcal {O}}(-n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a114380e72c0f6b7c526521795a3d7d70cf58808)
this can be given by the quotient variety
[1]: 24 ![{\displaystyle {\mathcal {O}}\oplus {\mathcal {O}}(-n)=(\mathbb {C} ^{2}-\{0\})\times \mathbb {C} ^{2}/\mathbb {C} ^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40b0f2540a5469eb0f67999057e0768cf3c0093c)
where the action of
![{\displaystyle \mathbb {C} ^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12e4a3fc8042b1c73d31cfe579c92d34f1255137)
is given by
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f7bd1770b03977a3179032c0c07f7b8559d65de)
Then, the projectivization
![{\displaystyle \mathbb {P} ({\mathcal {O}}\oplus {\mathcal {O}}(-n))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a1b4e79721e214a495f4c025678343740b4f7d3)
is given by another
![{\displaystyle \mathbb {C} ^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12e4a3fc8042b1c73d31cfe579c92d34f1255137)
-action
[1]: 22 sending an equivalence class
![{\displaystyle [l_{0},l_{1},t_{0},t_{1}]\in {\mathcal {O}}\oplus {\mathcal {O}}(-n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/282b274df6d0bcfbddc53dfb8fbb15d1ff54c43b)
to
![{\displaystyle \mu \cdot [l_{0},l_{1},t_{0},t_{1}]=[l_{0},l_{1},\mu t_{0},\mu t_{1}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b23c5446f6d7dc59c268f6ff0550ce4dd395bb)
Combining these two actions gives the original quotient up top.
Transition maps
One way to construct this
-bundle is by using transition functions. Since affine vector bundles are necessarily trivial, over the charts
of
defined by
there is the local model of the bundle
![{\displaystyle U_{i}\times \mathbb {P} ^{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/715e81bea340d5640d326a9904763203055df7dd)
Then, the transition maps, induced from the transition maps of
![{\displaystyle {\mathcal {O}}\oplus {\mathcal {O}}(-n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a114380e72c0f6b7c526521795a3d7d70cf58808)
give the map
![{\displaystyle U_{0}\times \mathbb {P} ^{1}|_{U_{1}}\to U_{1}\times \mathbb {P} ^{1}|_{U_{0}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2506af1c76857ab57c7be2f1e8116e5d3a864c05)
sending
![{\displaystyle (X_{0},[y_{0}:y_{1}])\mapsto (X_{1},[y_{0}:x_{0}^{n}y_{1}])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79efd976f8367afc6a419abaf949a796270dee95)
where
![{\displaystyle X_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d)
is the affine coordinate function on
![{\displaystyle U_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b21a6f475b0e68475c6019abe1fed0b415e0e42)
.
[2] Properties
Projective rank 2 bundles over P1
Note that by Grothendieck's theorem, for any rank 2 vector bundle
on
there are numbers
such that
![{\displaystyle E\cong {\mathcal {O}}(a)\oplus {\mathcal {O}}(b).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8818b4b16fd75e2d1bb1afca121370039db3f366)
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66a57864e96e43197bf79f75e6ad71b5a153733d)
is the Hirzebruch surface
![{\displaystyle \Sigma _{b-a}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5725147143117807bcecf154035778fab770b338)
since this bundle can be tensored by
![{\displaystyle {\mathcal {O}}(-a)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22e1161778c03e4ada6d7341b9182279b8cfa994)
.
Isomorphisms of Hirzebruch surfaces
In particular, the above observation gives an isomorphism between
and
since there is the isomorphism vector bundles
![{\displaystyle {\mathcal {O}}(n)\otimes ({\mathcal {O}}\oplus {\mathcal {O}}(-n))\cong {\mathcal {O}}(n)\oplus {\mathcal {O}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a63c872f8429ce738ed2878ef9cb5f9ea6a2fa7e)
Analysis of associated symmetric algebra
Recall that projective bundles can be constructed using Relative Proj, which is formed from the graded sheaf of algebras
![{\displaystyle \bigoplus _{i=0}^{\infty }\operatorname {Sym} ^{i}({\mathcal {O}}\oplus {\mathcal {O}}(-n))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8b16e2853d6a16bdd2f19ce681355e770d64ff3)
The first few symmetric modules are special since there is a non-trivial anti-symmetric
![{\displaystyle \operatorname {Alt} ^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/614e2f20f81f5bc3f448dc6eca1464761b05c3f1)
-module
![{\displaystyle {\mathcal {O}}\otimes {\mathcal {O}}(-n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/760e00b593b93e8bdf9e9d5945e680a93571baa7)
. These sheaves are summarized in the table
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8466de96b8a7e854570bafa73220e3a6b7c93ab)
For
![{\displaystyle i>2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73705abb8c5d4b89d3a193548d3eda3cefe85496)
the symmetric sheaves are given by
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5dd1e8700f12763b0f2afa234028f7630461752)
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
![{\displaystyle {\begin{bmatrix}0&1\\1&-n\end{bmatrix}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23f504494119561d0505e8c9bc9d03115043ab38)
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
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.
- 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