Introduction

      Key Warning and Disclaimer

      Overview of the Chapter

 
Hilbert Series and Graded Rings

      Hilbert Series and Hilbert Polynomials
            HilbertFunction(p,V) : RngUPolElt, SeqEnum -> UserProgram
            HilbertSeries(p,V) : RngUPolElt, SeqEnum -> FldFunRatUElt

      Interpreting the Hilbert Numerator
            HilbertSeriesMultipliedByMinimalDenominator(p,V) : RngUPolElt, SeqEnum -> RngUPolElt, SeqEnum
            HilbertNumerator(g, D) : FldFunRatUElt, SeqEnum -> FldFunRatUElt
            Example GrdRng_gr-genus4curve (H128E1)
            FindFirstGenerators(g) : FldFunRatUElt -> SeqEnum
            Example GrdRng_gr-grfirstgens (H128E2)
            ApparentCodimension(f) : RngUPolElt -> RngIntElt

 
Baskets of Singularities

      Point Singularities
            Example GrdRng_gr-grpoints (H128E3)

            Creation of Point Singularities
                  Point(r,n,Q) : RngIntElt, RngIntElt, SeqEnum -> GRPtS

            Accessing the Key Data and Testing Equality
                  Dimension(p) : GRPtS -> RngIntElt
                  Index(p) : GRPtS -> RngIntElt
                  Polarisation(p) : GRPtS -> SeqEnum
                  Eigenspace(p) : GRPtS -> RngIntElt
                  p eq q : GRPtS, GRPtS -> BoolElt

            Identifying Special Types of Point Singularity
                  IsIsolated(p) : GRPtS -> BoolElt
                  IsGorensteinSurface(p) : GRPtS -> BoolElt
                  IsTerminalThreefold(p) : GRPtS -> BoolElt
                  TerminalIndex(p) : GRPtS -> RngIntElt
                  TerminalPolarisation(p) : GRPtS -> SeqEnum
                  IsCanonical(p) : GRPtS -> BoolElt

      Curve Singularities
            Example GrdRng_gr-curvesing (H128E4)

            Creation of Curve Singularities
                  Curve(d,p,m) : FldRatElt,GRPtS,FldRatElt -> GRCrvS

            Accessing the Key Data and Testing Equality
                  Degree(C) : GRCrvS -> RngIntElt
                  TransverseType(C) : GRCrvS -> GRPtS
                  TransverseIndex(C) : GRCrvS -> RngIntElt
                  NormalNumber(C) : GRCrvS -> RngIntElt
                  Index(C) : GRCrvS -> RngIntElt
                  MagicNumber(C) : GRCrvS -> RngIntElt
                  Dimension(C) : GRCrvS -> RngIntElt
                  IsCanonical(C) : GRCrvS -> BoolElt
                  C eq D : GRCrvS, GRCrvS -> BoolElt

      Baskets of Singularities

            Creation and Modification of Baskets
                  Basket(Q) : SeqEnum -> GRBskt
                  EmptyBasket() : . -> GRBskt
                  MakeBasket(Q) : SeqEnum -> GRBskt
                  Points(B) : GRBskt -> SeqEnum
                  Curves(B) : GRBskt -> SeqEnum

            Tests for Baskets
                  IsIsolated(B) : GRBskt -> BoolElt
                  IsGorensteinSurface(B) : GRBskt -> BoolElt
                  IsTerminalThreefold(B) : GRBskt -> BoolElt
                  IsCanonical(B) : GRBskt -> BoolElt

      Curves and Dissident Points
            CanonicalDissidentPoints(C) : GRCrvS -> SeqEnum
            SimpleCanonicalDissidentPoints(C) : GRCrvS -> SeqEnum
            PossibleCanonicalDissidentPoints(C) : GRCrvS -> SeqEnum
            PossibleSimpleCanonicalDissidentPoints(C) : GRCrvS -> SeqEnum

 
Generic Polarised Varieties
      PolarisedVariety(d,W,n) : RngIntElt,SeqEnum,RngUPolElt-> GRSch

      Accessing the Data
            Weights(X) : GRSch -> SeqEnum
            Degree(X) : GRSch -> FldRatElt
            Basket(X) : GRSch -> Bskt
            RawBasket(X) : GRSch -> SeqEnum
            Dimension(X) : GRSch -> RngIntElt
            Codimension(X) : GRSch -> RngIntElt
            HilbertNumerator(X) : GRSch -> RngUPolElt
            NoetherWeights(X) : GRSch -> SeqEnum
            NoetherNumerator(X) : GRSch -> RngUPolElt
            NoetherNormalisation(X) : GRSch -> Tup
            HilbertSeries(X) : GRSch -> FldFunRatUElt
            InitialCoefficients(X) : GRSch -> SeqEnum
            ApparentCodimension(X) : GRSch -> RngIntElt

      Generic Creation, Checking, Changing
            X eq Y : GRSch,GRSch -> BoolElt
            CheckCodimension(X) : GRSch -> BoolElt
            FirstWeights(X) : GRSch -> SeqEnum
            IncludeWeight(~X,w) : GRSch,RngIntElt ->
            RemoveWeight(~X,w) : GRSch,RngIntElt ->
            MinimiseWeights(~X) : GRSch ->

 
Subcanonical Curves

      Creation of Subcanonical Curves
            SubcanonicalCurve(g,d,Q) : RngIntElt,RngIntElt,SeqEnum -> GRCrvK
            IsSubcanonicalCurve(g,d,Q) : RngIntElt,RngIntElt,SeqEnum -> BoolElt,GRCrvK
            HilbertPolynomialOfCurve(g,m) : RngIntElt,RngIntElt -> RngUPolElt
            IsEffective(C) : GRCrvK -> BoolElt

      Catalogue of Subcanonical Curves
            EffectiveSubcanonicalCurves(g) : RngIntElt -> SeqEnum
            IneffectiveSubcanonicalCurves(g) : RngIntElt -> SeqEnum

 
K3 Surfaces

      Creating and Comparing K3 Surfaces
            K3Surface(g,B) : RngIntElt,GRBskt -> GRK3
            K3Copy(X) : GRK3 -> GRK3

      Accessing the Key Data
            Genus(X) : GRK3 -> RngIntElt
            TwoGenus(X) : GRK3 -> RngIntElt
            SingularRank(X) : GRK3 -> RngIntElt
            AFRNumber(X) : GRK3 -> RngIntElt

      Modifying K3 Surfaces
            IncludeWeight(X,w) : GRK3,RngIntElt -> GRK3
            RemoveWeight(X,w) : GRK3,RngIntElt -> GRK3

 
Weil Polynomials
      SetVerbose("WeilPolynomials", v) : MonStgElt, RngIntElt ->
      HasAllRootsOnUnitCircle(f) : RngUPolElt -> BoolElt
      FrobeniusTracesToWeilPolynomials(tr, q, i, deg) : SeqEnum, RngIntElt, RngIntElt, RngIntElt -> SeqEnum
      WeilPolynomialToRankBound(f, q) : RngUPolElt, RngIntElt -> RngIntElt
      ArtinTateFormula(f, q, h20) : RngUPolElt, RngIntElt, RngIntElt -> RngIntElt, RngIntElt
      WeilPolynomialOverFieldExtension(f, deg) : RngUPolElt, RngIntElt -> RngUPolElt
      CheckWeilPolynomial(f, q, h20) : RngUPolElt, RngIntElt, RngIntElt -> BoolElt
      Example GrdRng_weilpoly (H128E5)
      Example GrdRng_weilpoly2 (H128E6)

 
Point Counting on Degree Two K3 Surfaces
      SetVerbose("Degree2K3", v) : MonStgElt, RngIntElt ->
      WeilPolynomialOfDegree2K3Surface(f6) : RngMPolElt -> RngUPolElt, RngUPolElt
      NonOrdinaryPrimes(f6,lim) : RngMPolElt, RngIntElt -> SeqEnum
      NumbersOfPointsOnDegree2K3Surface(f6,p,d) : RngMPolElt, RngIntElt, RngIntElt -> SeqEnum
      Example GrdRng_Degree2K3 (H128E7)

 
The K3 Database

      Searching the K3 Database
            Example GrdRng_k3db-ex1 (H128E8)
            K3Database() : -> DB
            Number(D,X) : DB,GRK3 -> RngIntElt,GRK3
            Index(D,X) : DB,GRK3 -> RngIntElt,GRK3
            Example GrdRng_gr-k3surface (H128E9)

      Working with the K3 Database
            K3Surface(D,i) : DB,RngIntElt -> GRK3
            K3Surface(D,Q,i) : DB,SeqEnum,RngIntElt -> GRK3
            K3Surface(D,g,i) : DB,RngIntElt,RngIntElt -> GRK3
            K3Surface(D,g1,g2,i) : DB,RngIntElt,RngIntElt,RngIntElt -> GRK3
            K3Surface(D,W) : DB,SeqEnum -> GRK3
            K3Surface(D,g,B) : DB,RngIntElt,GRBskt -> GRK3

 
Fano 3-folds
      Example GrdRng_gr-fano (H128E10)

      Creation: f=1, 2 or ≥3
            Fano(f,B,g) : RngIntElt,GRBskt,RngIntElt -> GRFano
            Fano(f,B) : RngIntElt,GRBskt -> GRFano
            FanoIndex(X) : GRFano -> RngIntElt
            FanoGenus(X) : GRFano -> RngIntElt
            FanoBaseGenus(X) : GRFano -> RngIntElt
            BogomolovNumber(X) : GRFano -> FldRatElt
            IsBogomolovUnstable(X) : GRFano -> BoolElt

      A Preliminary Fano Database
            FanoDatabase() : -> DB
            Fano(D,i) : DB,RngIntElt -> GRFano
            Fano(D,f,i) : DB,RngIntElt,RngIntElt -> GRFano
            Fano(D,f,Q,i) : DB,SeqEnum,RngIntElt -> GRFano

 
Calabi--Yau 3-folds
      CalabiYau(p1,p2,B) : RngIntElt,RngIntElt,GRBskt -> GRCY
      FindN(X) : GRCY -> RngIntElt,RngIntElt
      FindN(p1,p2,B) : RngIntElt,RngIntElt,GRBskt -> RngIntElt,RngIntElt

 
Building Databases

      The K3 Database

            Creating Many K3 Surfaces
                  CreateK3Data(g) : RngIntElt -> SeqEnum

            K3 Surfaces as Records
                  K3SurfaceToRecord(X) : GRK3 -> Rec
                  K3Surface(x) : Rec -> GRK3

            Writing K3 Surfaces to a File
                  WriteK3Data(Q,F) : SeqEnum,MonStgElt ->

            Writing the Data and Index Files

            Reading the Raw Data
                  K3SurfaceRaw(D,i) : DB,RngIntElt -> Tup
                  K3Surface(x) : Tup -> GRK3

      Making New Databases

 
Bibliography

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