Introduction

 
Generalities

      Ambients

      Surfaces

      Singularity

      Maps and Points

      Sheaves and Divisors

 
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 (H126E1)

      Invariants
            GeometricGenus(S) : Srfc -> RngIntElt
            Plurigenus(S,n) : Srfc, RngIntElt -> 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 (H126E2)

      Singularity Properties
            IsNormal(S) : Srfc -> BoolElt
            IsSimpleSurfaceSingularity(p) : Pt -> BoolElt, MonStr, RngIntElt
            HasOnlySimpleSingularities(S) : Srfc -> BoolElt, List
            Example AlgSrf_srfc_simp_sings (H126E3)

      Kodaira-Enriques Classification
            KodairaEnriquesType(S) : Srfc -> RngIntElt, RngIntElt, MonStr
            KodairaDimension(S) : Srfc -> RngIntElt
            Example AlgSrf_srfc-kod-enr-ex (H126E4)

      Minimal Models
            MinimalModelRationalSurface(S) : Srfc -> Map
            Example AlgSrf_min-mod-rat (H126E5)
            MinimalModelRuledSurface(S) : Srfc -> Map
            MinimalModelKodairaDimensionZero(S) : Srfc -> Map
            Example AlgSrf_srfcs_kod_dim_0_min_mods1 (H126E6)
            Example AlgSrf_srfcs_kod_dim_0_min_mods2 (H126E7)
            MinimalModelKodairaDimensionOne(S) : Srfc -> Map, Map
            MinimalModelGeneralType(S) : Srfc -> Map, BoolElt
            CanonicalCoordinateIdeal(S) : Srfc -> RngMPol
            Example AlgSrf_srfcs_gen_typ_min_mods1 (H126E8)
            Example AlgSrf_srfcs_gen_typ_min_mods2 (H126E9)

      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

 
Desingularisation by Blow Up

      Introduction
            ResolveSingularSurface(S) : Srfc -> List, RngIntElt
            DesingulariseSurfaceByBlowUp(S) : Srfc -> List
            ResolveSingByBlowUp(X,Y) : Sch, Sch -> DesingData

      Accessor Functions
            NumberOfBlowUpDivisors(dsd) : DesingData -> RngIntElt
            SingularPoint(dsd) : DesingData -> Sch
            BlowUpDivisor(S, dsd, i) : Srfc, DesingData, RngIntElt -> Crv, Sch, MapSch
            BlowUpDivisorAllPatches(dsd, i) : DesingData, RngIntElt -> List, RngIntElt
            Example AlgSrf_srfc_blow_up_desing_basics (H126E10)

      Multiplicities, Intersections and Restricted Linear Systems
            IntersectionMatrix(dsd) : DesingData -> Mtrx
            Multiplicities(S,D) : Srfc, DivSchElt -> SeqEnum
            MultiplicitiesAndIntersections(S,D) : Srfc, DivSchElt -> SeqEnum, SeqEnum
            IntersectionNumberOfStrictTransforms(S,D1,D2) : Srfc, Sch, Sch -> RngIntElt
            SelfIntersectionOfStrictTransform(S,D) : Srfc, Sch -> RngIntElt
            LinearSystemDivisorRestriction(S,B,ms) : Srfc, SeqEnum[RngMPolElt], SeqEnum[SeqEnum] -> SeqEnum

      Canonical Divisor Functionality
            DifferentialMultiplicities(dsd) : DesingData -> SeqEnum
            BasisOfHolomorphicTwoForms(S) : Srfc -> SeqEnum[RngMPolElt], RngMPolElt, SeqEnum, List
            PluriCanonicalBasis(S,m) : Srfc, RngIntElt -> SeqEnum
            Example AlgSrf_srfc_sing_dp4_2frms_n_pluri (H126E11)
            FirstChernClassOfDesingularization(S) : Srfc -> RngIntElt[RngMPolElt]
            CanonicalIntersection(S,D) : Srfc, DivSchElt -> RngIntElt
            ExceptionalDivisors(S) : Srfc -> List, List
            Example AlgSrf_srfc_ex_divs_intr (H126E12)

      Dual Resolution Graphs
            DualResolutionGraph(dsd) : DesingData -> GrphMultUnd, SeqEnum, SeqEnum
            MinimalDualResolutionGraph(dsd) : DesingData -> GrphMultUnd, SeqEnum, SeqEnum, Mtrx
            Example AlgSrf_srfc_dual_desing_grph_1 (H126E13)
            Example AlgSrf_srfc_dual_desing_grph_2 (H126E14)
            SelfIntersection(dsd,i) : DesingData,RngIntElt -> RngIntElt
            CanonicalIntersection(dsd,i) : DesingData,RngIntElt -> RngIntElt
            ArithmeticGenus(dsd,i) : DesingData,RngIntElt -> RngIntElt
            Genus(dsd,i) : DesingData,RngIntElt -> RngIntElt
            LocalIntersectionNumber(e) : GrphEdge -> RngIntElt

            Point counting on resolution fibres
                  NumberOfPointsOnResolutionFibre(dsd) : DesingData -> RngIntElt
                  NumberOfPointsOnMinimalResolutionFibre(dsd) : DesingData -> RngIntElt
                  ZetaFunctionOfResolutionFibre(dsd) : DesingData -> FldFunRatUElt
                  ZetaFunctionOfResolutionFibre(dsd) : DesingData -> FldFunRatUElt
                  ZetaFunctionOfMinimalResolutionFibre(dsd) : DesingData -> FldFunRatUElt
                  Example AlgSrf_srfc_dual_desing_grph_zeta (H126E15)

      Extended Examples
            Example AlgSrf_srfc_blow_up_desing_1 (H126E16)
            Example AlgSrf_srfc_blow_up_desing_2 (H126E17)
            Example AlgSrf_srfc_blow_up_desing_3 (H126E18)

 
Surfaces in P³

      Introduction

      Embedded Formal Desingularization of Curves
            ResolveAffineCurve(p) : RngMPolElt -> List, List, List, RngIntElt
            Example AlgSrf_aff_crv_res (H126E19)
            ResolveProjectiveCurve(p) : RngMPolElt -> List, List, List, RngIntElt
            Example AlgSrf_prj_crv_res (H126E20)

      Formal Desingularization of Surfaces
            ResolveAffineMonicSurface(s) : RngUPolElt -> List, RngIntElt
            Example AlgSrf_aff_res (H126E21)
            FormallyResolveProjectiveHyperSurface(S): Srfc -> List, RngIntElt
            Example AlgSrf_prj_res (H126E22)

      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 (H126E23)

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

      Reduction to Special Models
            ClassifyRationalSurface(S) : Srfc -> Srfc, List, MonStgElt
            Example AlgSrf_class (H126E24)

      Parametrization of Rational Surfaces
            ParametrizeProjectiveHypersurface(X, P2) : Srfc, Prj -> BoolElt, MapSch
            ParametrizeProjectiveSurface(X, P2) : Srfc, Prj -> BoolElt, MapSch
            Example AlgSrf_prm (H126E25)
            Solve(p, F) : RngMPolElt, FldFunRat -> SeqEnum
            Example AlgSrf_prm2 (H126E26)

      Parametrization of Special Surfaces
            ParametrizeQuadric(X,P2) : Sch, Prj -> BoolElt, MapSch
            Example AlgSrf_solve_quad (H126E27)
            ParametrizePencil(phi, P2) : MapSch, Prj -> BoolElt, MapSch
            Example AlgSrf_ruled_ex (H126E28)
            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

      Some Auxiliary Intrinsics
            CollinearPointsOnPlane(P2,S) : Prj,SetIndx[Pt] -> SeqEnum
            PointsInGeneralPosition(P2,S) : Prj,SetIndx[Pt] -> BoolElt,SeqEnum,SeqEnum,SeqEnum
            EckardtPoints(X) : SrfDelPezzo -> Sch

      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 (H126E29)
            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 (H126E30)
            ParametrizeDegree5DelPezzo(X) : Sch -> MapIsoSch
            ParametrizeSingularDegree3DelPezzo(X,P2) : Sch, Prj -> BoolElt, MapIsoSch
            Example AlgSrf_del-pezzo-ex-3-sing (H126E31)

      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 (H126E32)

      Cubic Surfaces over Finite Fields
            NumberOfPointsOnCubicSurface(f) : RngMPolElt -> RngIntElt, RngIntElt
            Example AlgSrf_numptscubic (H126E33)
            IsIsomorphicCubicSurface(f,g) : MPolElt, MPolElt -> BoolElt, List
            Example AlgSrf_isisocubic (H126E34)

      Construction of Cubic Surfaces
            CubicSurfaceByHexahedralCoefficients(p) : RngUPolElt -> RngMPolElt
            CoblesRadicand(p) : RngUPolElt -> FldElt
            Example AlgSrf_hex (H126E35)

      Invariant Theory of Cubic Surfaces

            Invariants
                  ClebschSalmonInvariants(f) : RngMPolElt -> SeqEnum, RngElt
                  SkewInvariant100(f) : RngMPolElt -> RngElt
                  CubicSurfaceFromClebschSalmon(inv) : SeqEnum -> RngMPolElt
                  Example AlgSrf_inv_cub (H126E36)

            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 (H126E37)

            Interaction of Covariants and Contravariants
                  ApplyContravariant(c, d) : MPolElt, MPolElt -> MPolElt
                  Example AlgSrf_co_cont (H126E38)

      The Pentahedron of a Cubic Surface
            PentahedronIdeal(f) : RngMPolElt -> RngMPol
            Example AlgSrf_penta (H126E39)

 
Degree 2 K3 Surfaces

      Introduction

      Creation Functions
            DegreeTwoK3Surface(f) : RngMPolElt -> Srfc
            Example AlgSrf_srfc_deg2K3_constr (H126E40)

      Split Divisors and Intersection Matrices
            TriTangentLines(S) : Srfc -> List
            SixTangentConicsModp(F) : RngMPolElt -> SeqEnum[RngMPolElt]
            SixTangentConicCheck(S,C) : Srfc, RngMPolElt -> BoolElt
            IntersectionMatrixOnDegree2K3(S, Ds) : Srfc, SeqEnum[Sch] -> Mtrx
            Example AlgSrf_deg2K3_int_pairings (H126E41)

      Elliptic Fibrations
            KodairaConfigurations(imat) : Mtrx -> List
            Example AlgSrf_deg2K3_int_pairings (H126E42)
            EllipticFibrationRRSpaceDeg2K3(S,divlst,exdivlst) : Srfc, SeqEnum, SeqEnum -> SeqEnum[RngMPolElt],RngMPolElt
            EllipticGeneralFibreDeg2K3(S,B,secs) : Srfc, SeqEnum[RngMPolElt], SeqEnum[Sch] -> Crv,SeqEnum,SeqEnum
            Example AlgSrf_deg2K3-ell-fib-ex (H126E43)

 
Bibliography

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