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ALGEBRAIC SURFACES

 
Acknowledgements
 
Introduction
 
General Surfaces
      Introduction
      Creation Functions
      Invariants
      Singularity Properties
      Kodaira-Enriques Classification
      Minimal Models
      Special Surfaces in Projective 4-space
 
Surfaces in Prj3
      Introduction
      Embedded Formal Desingularization of Curves
      Formal Desingularization of Surfaces
      Adjoint Systems and Birational Invariants
      Classification and Parameterization of Rational Surfaces
      Reduction to Special Models
      Parametrization of Rational Surfaces
      Parametrization of Special Surfaces
 
Del Pezzo Surfaces
      Introduction
      Creation of General Del Pezzos
      Parametrization of Del Pezzo Surfaces
      Minimization and Reduction of Surfaces
      Cubic Surfaces over Finite Fields
      Construction of Cubic Surfaces
      Invariant Theory of Cubic Surfaces
            Invariants
            Covariants
            Contravariants
            Interaction of Covariants and Contravariants
      The Pentahedron of a Cubic Surface
 
Bibliography







DETAILS

 
Introduction

 
General Surfaces

      Introduction

      Creation Functions
            Surface(A,I) : Sch, ModMPol -> Srfc
            RationalRuledSurface(P,n) : Prj, RngIntElt -> Srfc, MapSch
            RandomCompleteIntersection(P,ds) : Prj, SeqEnum[RngIntElt] -> Sch
            KummerSurfaceScheme(C) : CrvHyp -> Srfc
            Example AlgSrf_srfcs-creation (H116E1)

      Invariants
            GeometricGenus(S) : Srfc -> RngIntElt
            Plurigenus(S,n) : Srfc -> RngIntElt
            ArithmeticGenus(S) : Srfc -> RngIntElt
            Irregularity(S) : Srfc -> RngIntElt
            ChernNumber(S,n) : Srfc, RngIntElt -> RngIntElt
            MinimalChernNumber(S,n) : Srfc, RngIntElt -> RngIntElt
            HodgeNumber(S,i,j) : Srfc, RngIntElt, RngIntElt -> RngIntElt
            Example AlgSrf_srfcs-invs (H116E2)

      Singularity Properties
            IsNormal(S) : Srfc -> BoolElt
            IsSimpleSurfaceSingularity(p) : Pt -> BoolElt, MonStr, RngIntElt
            HasOnlySimpleSingularities(S) : Srfc -> BoolElt, List
            Example AlgSrf_srfc_simp_sings (H116E3)

      Kodaira-Enriques Classification
            KodairaEnriquesType(S) : Srfc -> RngIntElt, RngIntElt, MonStr
            KodairaEnriquesDimension(S) : Srfc -> RngIntElt
            Example AlgSrf_srfc-kod-enr-ex (H116E4)

      Minimal Models
            MinimalModelRationalSurface(S) : Srfc -> Map
            Example AlgSrf_min-mod-rat (H116E5)
            MinimalModelRuledSurface(S) : Srfc -> Map
            MinimalModelKodairaDimensionZero(S) : Srfc -> Map
            Example AlgSrf_srfcs_kod_dim_0_min_mods1 (H116E6)
            Example AlgSrf_srfcs_kod_dim_0_min_mods2 (H116E7)
            MinimalModelKodairaDimensionOne(S) : Srfc -> Map, Map
            MinimalModelGeneralType(S) : Srfc -> Map, BoolElt
            CanonicalCoordinateIdeal(S) : Srfc -> RngMPol
            Example AlgSrf_srfcs_gen_typ_min_mods1 (H116E8)
            Example AlgSrf_srfcs_gen_typ_min_mods2 (H116E9)

      Special Surfaces in Projective 4-space
            RandomRationalSurface_d10g9(P) : Prj -> Srfc
            RandomEnriquesSurface_d9g6(P) : Prj -> Srfc
            RandomAbelianSurface_d10g6(P) : Prj -> Srfc
            RandomEllipticFibration_d7g6(P) : Prj -> Srfc
            RandomEllipticFibration_d8g7(P) : Prj -> Srfc
            RandomEllipticFibration_d9g7(P) : Prj -> Srfc
            RandomEllipticFibration_d10g10(P) : Prj -> Srfc

 
Surfaces in Prj3

      Introduction

      Embedded Formal Desingularization of Curves
            ResolveAffineCurve(p) : RngMPolElt -> List, List, List, RngIntElt
            Example AlgSrf_aff_crv_res (H116E10)
            ResolveProjectiveCurve(p) : RngMPolElt -> List, List, List, RngIntElt
            Example AlgSrf_prj_crv_res (H116E11)

      Formal Desingularization of Surfaces
            ResolveAffineMonicSurface(s) : RngUPolElt -> List, RngIntElt
            Example AlgSrf_aff_res (H116E12)
            ResolveProjectiveSurface(S) : Srfc -> List, RngIntElt
            Example AlgSrf_prj_res (H116E13)

      Adjoint Systems and Birational Invariants
            HomAdjoints(m,n,S) : RngIntElt, RngIntElt, Srfc -> SeqEnum
            GeometricGenusOfDesingularization(S) : Srfc -> RngIntElt
            PlurigenusOfDesingularization(S,m) : Srfc, RngIntElt -> RngIntElt
            ArithmeticGenusOfDesingularization(S) : Srfc -> RngIntElt
            Example AlgSrf_adj_ex (H116E14)

      Classification and Parameterization of Rational Surfaces
            IsRational(X) : Srfc -> BoolElt

      Reduction to Special Models
            ClassifyRationalSurface(S) : Srfc -> Srfc, List, MonStgElt
            Example AlgSrf_class (H116E15)

      Parametrization of Rational Surfaces
            ParametrizeProjectiveHypersurface(X, P2) : Srfc, Prj -> BoolElt, MapSch
            ParametrizeProjectiveSurface(X, P2) : Srfc, Prj -> BoolElt, MapSch
            Example AlgSrf_prm (H116E16)
            Solve(p, F) : RngMPolElt, FldFunRat -> SeqEnum
            Example AlgSrf_prm2 (H116E17)

      Parametrization of Special Surfaces
            ParametrizeQuadric(X,P2) : Sch, Prj -> BoolElt, MapSch
            Example AlgSrf_solve_quad (H116E18)
            ParametrizePencil(phi, P2) : MapSch, Prj -> BoolElt, MapSch
            Example AlgSrf_ruled_ex (H116E19)
            ParametrizeDelPezzo(X, P2) : Sch, Prj -> BoolElt, MapSch

 
Del Pezzo Surfaces

      Introduction

      Creation of General Del Pezzos
            DelPezzoSurface(P,L) : Prj,List -> SrfDelPezzo
            DelPezzoSurface(f) : RngMPolElt -> SrfDelPezzo
            IsDelPezzo(Y) : Sch -> BoolElt, SrfDelPezzo, MapSch

      Parametrization of Del Pezzo Surfaces
            SetVerbose("ParamDP", v) : MonStgElt, RngIntElt ->
            ParametrizeDegree9DelPezzo(X) : Sch -> BoolElt, MapIsoSch
            ParametrizeDegree8DelPezzo(X) : Sch -> BoolElt, MapSch
            Example AlgSrf_del-pezzo-ex-8 (H116E20)
            ParametrizeDegree7DelPezzo(X) : Sch -> MapIsoSch
            ParametrizeDegree6DelPezzo(X) : Sch -> BoolElt, MapIsoSch
            Degree6DelPezzoType2_1(K,pt) : FldNum, Pt -> Sch
            ParametrizeDelPezzoDeg6(X) : Sch -> BoolElt, MapIsoSch
            Example AlgSrf_del-pezzo-ex-6 (H116E21)
            ParametrizeDegree5DelPezzo(X) : Sch -> MapIsoSch
            ParametrizeSingularDegree3DelPezzo(X,P2) : Sch, Prj -> BoolElt, MapIsoSch
            Example AlgSrf_del-pezzo-ex-3-sing (H116E22)

      Minimization and Reduction of Surfaces
            MinimizeCubicSurface(f, p) : RngMPolElt, RngIntElt -> RngMPolElt, Mtrx
            ReduceCubicSurface(f) : RngMPolElt -> RngMPolElt, Mtrx
            MinimizeReduceCubicSurface(f) : MPolElt -> RngMPolElt, Mtrx
            MinimizeDeg4delPezzo(f, p) : SeqEnum, RngIntElt -> SeqEnum, Mtrx
            MinimizeReduceDeg4delPezzo(f) : SeqEnum -> SeqEnum, Mtrx
            MinimizeReduce(S) : SrfDelPezzo -> SrfDelPezzo, Mtrx
            Example AlgSrf_dp34 (H116E23)

      Cubic Surfaces over Finite Fields
            NumberOfPointsOnCubicSurface(f) : RngMPolElt -> RngIntElt, RngIntElt
            Example AlgSrf_numptscubic (H116E24)
            IsIsomorphicCubicSurface(f,g) : MPolElt, MPolElt -> BoolElt, List
            Example AlgSrf_isisocubic (H116E25)

      Construction of Cubic Surfaces
            CubicSurfaceByHexahedralCoefficients(p) : RngUPolElt -> RngMPolElt
            CoblesRadicand(p) : RngUPolElt -> FldElt
            Example AlgSrf_hex (H116E26)

      Invariant Theory of Cubic Surfaces

            Invariants
                  ClebschSalmonInvariants(f) : RngMPolElt -> SeqEnum, RngElt
                  SkewInvariant100(f) : RngMPolElt -> RngElt
                  CubicSurfaceFromClebschSalmon(inv) : SeqEnum -> RngMPolElt
                  Example AlgSrf_inv_cub (H116E27)

            Covariants
                  LinearCovariants(f) : RngMPolElt -> SeqEnum
                  ClassicalCovariantsOfCubicSurface(f) : RngMPolElt -> SeqEnum

            Contravariants
                  NumericClebschTransfer(f, inv, p) : RngMPolElt, UserProgram, SeqEnum -> RngElt
                  ContravariantsOfCubicSurface(f) : RngMPolElt -> SeqEnum
                  Example AlgSrf_cayley_dual (H116E28)

            Interaction of Covariants and Contravariants
                  ApplyContravariant(c, d) : MPolElt, MPolElt -> MPolElt
                  Example AlgSrf_co_cont (H116E29)

      The Pentahedron of a Cubic Surface
            PentahedronIdeal(f) : RngMPolElt -> RngMPol
            Example AlgSrf_penta (H116E30)

 
Bibliography

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013