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gptkbp:instanceOf
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gptkb:algebraic_geometry
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gptkbp:anticanonicalBundle
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ample
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gptkbp:anticanonicalMap
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2-to-1 for degree 2
birational for degree ≥ 3
degree 1 for degree 1
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gptkbp:automorphismGroup
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finite for degree ≤ 5
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gptkbp:class
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classified by degree
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gptkbp:defines
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A smooth projective algebraic surface with ample anticanonical bundle
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gptkbp:degree
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integer between 1 and 9
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gptkbp:degree1
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blow-up of projective plane at 8 points
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gptkbp:degree2
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double cover of projective plane branched over quartic curve
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gptkbp:degree3
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cubic surface in projective 3-space
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gptkbp:degree4
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intersection of two quadrics in projective 4-space
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gptkbp:degree5
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blow-up of projective plane at 4 points
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gptkbp:degree6
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blow-up of projective plane at 3 points
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gptkbp:degree7
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blow-up of projective plane at 2 points
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gptkbp:degree8
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blow-up of projective plane at 1 point
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gptkbp:degree9
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gptkb:projective_plane
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gptkbp:dimensions
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2
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gptkbp:example
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product of two projective lines (degree 8)
projective plane blown up at up to 8 points in general position
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gptkbp:field
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gptkb:algebraic_geometry
algebraically closed field
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gptkbp:importantFor
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classification of algebraic surfaces
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gptkbp:linesOnDegree3
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27
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gptkbp:linesOnDegree4
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16
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gptkbp:linesOnDegree5
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10
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gptkbp:linesOnDegree6
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6
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gptkbp:linesOnDegree7
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3
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gptkbp:linesOnDegree8
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1
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gptkbp:linesOnDegree9
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0
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gptkbp:linesOnSurface
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finite number depending on degree
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gptkbp:namedAfter
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gptkb:Pasquale_del_Pezzo
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gptkbp:PicardNumber
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1 for generic Del Pezzo surface
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gptkbp:relatedTo
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gptkb:Weyl_group
gptkb:Fano_variety
gptkb:K3_surface
gptkb:Mori_theory
gptkb:Enriques_surface
gptkb:Galois_cohomology
gptkb:Exceptional_Lie_algebras
gptkb:Rational_surface
gptkb:Cubic_surface
gptkb:Root_system
Birational geometry
Minimal model program
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gptkbp:singularDelPezzoSurface
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possible if points are not in general position
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gptkbp:studiedBy
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gptkb:Pasquale_del_Pezzo
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gptkbp:bfsParent
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gptkb:Fano_manifold
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gptkbp:bfsLayer
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7
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https://www.w3.org/2000/01/rdf-schema#label
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Del Pezzo surface
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