Hirzebruch Surface

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

Definition

The Hirzebruch surface $\Sigma_n$ is the $\mathbb^1$-bundle (a projective bundle) over the projective line $\mathbb^1$, associated to the sheaf$\mathcal\oplus \mathcal(-n).$The notation here means: $\mathcal(n)$ is the -th tensor power of the Serre twist sheaf $\mathcal(1)$, 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: $\Sigma_n = (\Complex^2-\)\times (\Complex^2-\)/(\Complex^*\times\Complex^*)$ where the action of $\Complex^*\times\Complex^*$ is given by $(\lambda, \mu)\cdot(l_0,l_1,t_0,t_1) = (\lambda l_0, \lambda l_1, \mu t_0,\lambda^\mu t_1)\ .$ This action can be interpreted as the action of $\lambda$ on the first two factors comes from the action of $\Complex^*$ on $\Complex^2 - \$ defining $\mathbb^1$, and the second action is a combination of the construction of a direct sum of line bundles on $\mathbb^1$ and their projectivization. For the direct sum $\mathcal\oplus \mathcal(-n)$ this can be given by the quotient variety$\mathcal\oplus \mathcal(-n) = (\Complex^2-\)\times \Complex^2/\Complex^*$where the action of $\Complex^*$ is given by$\lambda \cdot (l_0,l_1,t_0,t_1) = (\lambda l_0, \lambda l_1,\lambda^0 t_0=t_0, \lambda^ t_1)$Then, the projectivization $\mathbb(\mathcal\oplus\mathcal(-n))$ is given by another $\Complex^*$-action sending an equivalence class _0,l_1,t_0,t_1\in\mathcal\oplus\mathcal(-n) to\mu \cdot _0,l_1,t_0,t_1= _0,l_1,\mu t_0,\mu t_1/math>Combining these two actions gives the original quotient up top.

Transition maps

One way to construct this $\mathbb^1$-bundle is by using transition functions. Since affine vector bundles are necessarily trivial, over the charts $U_0,U_1$ of $\mathbb^1$ defined by $x_i \neq 0$ there is the local model of the bundle$U_i\times \mathbb^1$Then, the transition maps, induced from the transition maps of $\mathcal\oplus \mathcal(-n)$ give the map$U_0\times\mathbb^1, _ \to U_1\times\mathbb^1, _$sending(X_0, _0:y_1 \mapsto (X_1, _0:x_0^n y_1where $X_i$ is the affine coordinate function on $U_i$.

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$E \cong \mathcal(a)\oplus \mathcal(b).$As taking the projective bundle is invariant under tensoring by a line bundle, the ruled surface associated to $E = \mathcal O(a) \oplus \mathcal O(b)$ is the Hirzebruch surface $\Sigma_$ since this bundle can be tensored by $\mathcal(-a)$.

Isomorphisms of Hirzebruch surfaces

In particular, the above observation gives an isomorphism between $\Sigma_n$ and $\Sigma_$ since there is the isomorphism vector bundles$\mathcal(n)\otimes(\mathcal \oplus \mathcal(-n)) \cong \mathcal(n) \oplus \mathcal$

Analysis of associated symmetric algebra

Recall that projective bundles can be constructed using Relative Proj, which is formed from the graded sheaf of algebras$\bigoplus_^\infty \operatorname^i(\mathcal\oplus \mathcal(-n))$The first few symmetric modules are special since there is a non-trivial anti-symmetric $\operatorname^2$-module $\mathcal\otimes \mathcal(-n)$. These sheaves are summarized in the table$\begin \operatorname^0(\mathcal\oplus \mathcal(-n)) &= \mathcal \\ \operatorname^1(\mathcal\oplus \mathcal(-n)) &= \mathcal \oplus \mathcal(-n) \\ \operatorname^2(\mathcal\oplus \mathcal(-n)) &= \mathcal \oplus \mathcal(-2n) \end$For $i > 2$ the symmetric sheaves are given by$\begin \operatorname^k(\mathcal\oplus \mathcal(-n)) &= \bigoplus_^k \mathcal^\otimes \mathcal(-in) \\ &\cong \mathcal\oplus \mathcal(-n) \oplus \cdots \oplus \mathcal(-kn) \end$

Intersection theory

Hirzebruch surfaces for have a special rational curve on them: The surface is the projective bundle of $\mathcal(-n)$ and the curve is the zero section. This curve has self-intersection number , 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 and one of the fibers, and these generators have intersection matrix$\begin0 & 1 \\ 1 & -n \end ,$so the bilinear form is two dimensional unimodular, and is even or odd depending on whether is even or odd.
The Hirzebruch surface () blown up at a point on the special curve is isomorphic to 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 $T = \mathbb^*\times \mathbb^*$, with one $\mathbb^*$ acting on the base $\mathbb^1$ with two fixed axis points, and the other $\mathbb^*$ acting on the fibers of the vector bundle $\mathcal\oplus \mathcal(-n)$, 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 $\mathbb^2$ into four cones (each corresponding to a coordinate chart), separated by the rays along the four vectors:

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

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 $\mathbb^2$ can be constructed by repeatedly blowing up a Hirzebruch surface at ''T''-fixed points.

See also

* Projective bundle

References

*
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*{{Citation, last1=Hirzebruch , first1=Friedrich , author1-link=Friedrich Hirzebruch , title=Über eine Klasse von einfachzusammenhängenden komplexen Mannigfaltigkeiten , doi=10.1007/BF01343552 , mr=0045384 , year=1951 , journal=Mathematische Annalen , issn=0025-5831 , volume=124 , pages=77–86, hdl=21.11116/0000-0004-3A56-B , s2cid=122844063 , hdl-access=free

External links

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

Algebraic surfaces
Complex surfaces