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Hirzebruch surface

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

Definition

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The Hirzebruch surface is the -bundle (a projective bundle) over the projective line , associated to the sheafThe notation here means: is the n-th tensor power of the Serre twist sheaf , the invertible sheaf or line bundle with associated Cartier divisor a single point. The surface is isomorphic to ; and is isomorphic to the projective plane blown up at a point, so it is not minimal.

GIT quotient

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One method for constructing the Hirzebruch surface is by using a GIT quotient[1]: 21 where the action of is given byThis action can be interpreted as the action of on the first two factors comes from the action of on defining , and the second action is a combination of the construction of a direct sum of line bundles on and their projectivization. For the direct sum this can be given by the quotient variety[1]: 24 where the action of is given byThen, the projectivization is given by another -action[1]: 22  sending an equivalence class toCombining these two actions gives the original quotient up top.

Transition maps

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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 bundleThen, the transition maps, induced from the transition maps of give the mapsendingwhere is the affine coordinate function on .[2]

Properties

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Projective rank 2 bundles over P1

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Note that by Grothendieck's theorem, for any rank 2 vector bundle on there are numbers such thatAs taking the projective bundle is invariant under tensoring by a line bundle,[3] the ruled surface associated to is the Hirzebruch surface since this bundle can be tensored by .

Isomorphisms of Hirzebruch surfaces

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In particular, the above observation gives an isomorphism between and since there is the isomorphism vector bundles

Analysis of associated symmetric algebra

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Recall that projective bundles can be constructed using Relative Proj, which is formed from the graded sheaf of algebrasThe first few symmetric modules are special since there is a non-trivial anti-symmetric -module . These sheaves are summarized in the tableFor the symmetric sheaves are given by

Intersection theory

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Hirzebruch surfaces for n > 0 have a special rational curve C on them: The surface is the projective bundle of 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 ). The Picard group is generated by the curve C and one of the fibers, and these generators have intersection matrixso 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

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The Hirzebruch surface can be given an action of the complex torus , with one acting on the base with two fixed axis points, and the other acting on the fibers of the vector bundle , specifically on the first line bundle component, and hence on the projective bundle. This produces an open orbit of T, making a toric variety. Its associated fan partitions the standard lattice into four cones (each corresponding to a coordinate chart), separated by the rays along the four vectors:[4]

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 can be constructed by repeatedly blowing up a Hirzebruch surface at T-fixed points.[5]

See also

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References

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  1. ^ a b c 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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