- Source: 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
Σ
n
{\displaystyle \Sigma _{n}}
is the
P
1
{\displaystyle \mathbb {P} ^{1}}
-bundle (a projective bundle) over the projective line
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
P
1
×
P
1
{\displaystyle \mathbb {P} ^{1}\times \mathbb {P} ^{1}}
; and
Σ
1
{\displaystyle \Sigma _{1}}
is isomorphic to the projective plane
P
2
{\displaystyle \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: 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: 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
0
t
0
=
t
0
,
λ
−
n
t
1
)
{\displaystyle \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
P
(
O
⊕
O
(
−
n
)
)
{\displaystyle \mathbb {P} ({\mathcal {O}}\oplus {\mathcal {O}}(-n))}
is given by another
C
∗
{\displaystyle \mathbb {C} ^{*}}
-action: 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}}
.
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, 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
)
{\displaystyle {\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
P
1
{\displaystyle \mathbb {P} ^{1}}
). 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.
= Toric variety
=The Hirzebruch surface
Σ
n
{\displaystyle \Sigma _{n}}
can be given an action of the complex torus
T
=
C
∗
×
C
∗
{\displaystyle T=\mathbb {C} ^{*}\times \mathbb {C} ^{*}}
, with one
C
∗
{\displaystyle \mathbb {C} ^{*}}
acting on the base
P
1
{\displaystyle \mathbb {P} ^{1}}
with two fixed axis points, and the other
C
∗
{\displaystyle \mathbb {C} ^{*}}
acting on the fibers of the vector bundle
O
⊕
O
(
−
n
)
{\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
Σ
n
{\displaystyle \Sigma _{n}}
a toric variety. Its associated fan partitions the standard lattice
Z
2
{\displaystyle \mathbb {Z} ^{2}}
into four cones (each corresponding to a coordinate chart), separated by the rays along the four vectors:
(
1
,
0
)
,
(
0
,
1
)
,
(
0
,
−
1
)
,
(
−
1
,
n
)
.
{\displaystyle (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
P
2
{\displaystyle \mathbb {P} ^{2}}
can be constructed by repeatedly blowing up a Hirzebruch surface at T-fixed points.
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