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| In mathematics, a '''Hirzebruch surface''' is a [[ruled surface]] over the [[projective line]]. They were studied by {{harvs|txt|first=Friedrich|last=Hirzebruch|year=1951|authorlink=Friedrich Hirzebruch}}.
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| ==Definition==
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| The Hirzebruch surface
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| Σ<sub>n</sub> is the '''P'''<sup>1</sup> bundle over '''P'''<sup>1</sup>
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| associated to the [[Sheaf (mathematics)|sheaf]]
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| :<math>O(0)+O(-n).\ </math> | |
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| The notation here means: O(''n'') is the ''n''-th tensor power of the [[Serre twist sheaf]] O(1), the [[invertible sheaf]] or [[line bundle]] with associated [[Cartier divisor]] a single point. The surface Σ<sub>0</sub> is isomorphic to '''P'''<sup>1</sup>×'''P'''<sup>1</sup>, and Σ<sub>1</sub> is isomorphic to '''P'''<sup>2</sup> blown up at a point so is not minimal.
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| ==Properties==
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| Hirzebruch surfaces for ''n''>0 have a special [[rational curve]] ''C'' on them: The surface is
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| the projective bundle of O(''-n'') and the curve ''C'' is the [[zero section]]. This curve has [[Intersection theory|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'''<sup>1</sup>). The [[Picard group]] is generated by the curve ''C'' and one of the fibers, and these generators have intersection matrix
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| :<math>\begin{bmatrix}0 & 1 \\ 1 & -n \end{bmatrix} , </math> | |
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| so the bilinear form is two dimensional unimodular, and is even or odd depending on whether ''n'' is even or odd.
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| The Hirzebruch surface Σ<sub>''n''</sub> (''n'' > 1) blown up at a point on the special curve ''C'' is isomorphic to Σ<sub>n-1</sub> blown up at a point not on the special curve.
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| == External links ==
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| * [http://www.map.him.uni-bonn.de/index.php/Hirzebruch_surfaces Manifold Atlas]
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| === References ===
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| *{{Citation | last1=Barth | first1=Wolf P. | last2=Hulek | first2=Klaus | last3=Peters | first3=Chris A.M. | last4=Van de Ven | first4=Antonius | title=Compact Complex Surfaces | publisher= Springer-Verlag, Berlin | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | isbn=978-3-540-00832-3 | id={{MathSciNet | id = 2030225}} | year=2004 | volume=4}}
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| *{{Citation | last1=Beauville | first1=Arnaud | title=Complex algebraic surfaces | publisher=[[Cambridge University Press]] | edition=2nd | series=London Mathematical Society Student Texts | isbn=978-0-521-49510-3; 978-0-521-49842-5 | id={{MathSciNet | id = 1406314}} | year=1996 | volume=34}}
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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 | id={{MathSciNet | id = 0045384}} | year=1951 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=124 | pages=77–86}}
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| [[Category:Algebraic surfaces]]
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| [[Category:Complex surfaces]]
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