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complex surface is rational, because if a complex surface is unirational then its irregularity and plurigenera are bounded by those of a rational surface and are therefore all 0, so the surface is rational. Most unirational complex varieties of dimension 3 or larger are not rational. In
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Nonsingular cubic surfaces are isomorphic to the projective plane blown up in 6 points, and are Fano surfaces. Named examples include the
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of dimension two. Rational surfaces are the simplest of the 10 or so classes of surface in the
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Surface birationally equivalent to the projective plane; rational variety of dimension two
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An intersection of two quadrics, isomorphic to the projective plane blown up in 5 points.
547:: A tool to visually study the geography of (minimal) complex algebraic smooth surfaces
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proved that
Castelnuovo's theorem also holds over fields of positive characteristic.
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Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004),
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of complex surfaces, and were the first surfaces to be investigated.
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Every non-singular rational surface can be obtained by repeatedly
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At one time it was unclear whether a complex surface such that
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with singularities which is birational to the projective plane.
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The product of two projective lines is the
Hirzebruch surface ÎŁ
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defined by the quartics through 10 points in general position.
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314:: A degree 6 embedding of the projective plane into
502:(1958), "On Castelnuovo's criterion of rationality p
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both vanish is rational, but a counterexample (an
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263:Castelnuovo's theorem also implies that any
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512:Illinois Journal of Mathematics
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510:= 0 of an algebraic surface",
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306:Examples of rational surfaces
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475:Cambridge University Press
471:Complex algebraic surfaces
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431:List of algebraic surfaces
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545:Le Superficie Algebriche
445:Compact Complex Surfaces
344:Clebsch diagonal surface
72:minimal rational surface
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44:birationally equivalent
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238:Castelnuovo's theorem
340:Cayley cubic surface
566:Birational geometry
361:Hirzebruch surfaces
221:Hirzebruch surfaces
199:Hirzebruch surfaces
76:Hirzebruch surfaces
571:Algebraic surfaces
350:del Pezzo surfaces
210:unimodular lattice
102:are all 0 and the
32:algebraic geometry
484:978-0-521-49510-3
454:978-3-540-00832-3
322:Châtelet surfaces
300:Federigo Enriques
243:Guido Castelnuovo
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278:Zariski surfaces
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518:: 303–315,
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36:mathematics
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62:Structure
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