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HILBERT SERIES OF POLARISED VARIETIES

 
Acknowledgements
 
Introduction
      Key Warning and Disclaimer
      Overview of the Chapter
 
Hilbert Series and Graded Rings
      Hilbert Series and Hilbert Polynomials
      Interpreting the Hilbert Numerator
 
Baskets of Singularities
      Point Singularities
            Creation of Point Singularities
            Accessing the Key Data and Testing Equality
            Identifying Special Types of Point Singularity
      Curve Singularities
            Creation of Curve Singularities
            Accessing the Key Data and Testing Equality
      Baskets of Singularities
            Creation and Modification of Baskets
            Tests for Baskets
      Curves and Dissident Points
 
Generic Polarised Varieties
      Accessing the Data
      Generic Creation, Checking, Changing
 
Subcanonical Curves
      Creation of Subcanonical Curves
      Catalogue of Subcanonical Curves
 
K3 Surfaces
      Creating and Comparing K3 Surfaces
      Accessing the Key Data
      Modifying K3 Surfaces
 
The K3 Database
      Searching the K3 Database
      Working with the K3 Database
 
Fano 3-folds
      Creation: f=1, 2 or ≥3
      A Preliminary Fano Database
 
Calabi--Yau 3-folds
 
Building Databases
      The K3 Database
            Creating Many K3 Surfaces
            K3 Surfaces as Records
            Writing K3 Surfaces to a File
            Writing the Data and Index Files
            Reading the Raw Data
      Making New Databases
 
Bibliography







DETAILS

 
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 (H117E1)
            FindFirstGenerators(g) : FldFunRatUElt -> SeqEnum
            Example GrdRng_gr-grfirstgens (H117E2)
            ApparentCodimension(f) : RngUPolElt -> RngIntElt

 
Baskets of Singularities

      Point Singularities
            Example GrdRng_gr-grpoints (H117E3)

            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 (H117E4)

            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

 
The K3 Database

      Searching the K3 Database
            Example GrdRng_k3db-ex1 (H117E5)
            K3Database() : -> DB
            Number(D,X) : DB,GRK3 -> RngIntElt,GRK3
            Index(D,X) : DB,GRK3 -> RngIntElt,GRK3
            Example GrdRng_gr-k3surface (H117E6)

      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 (H117E7)

      Creation: f=1, 2 or ≥3
            Fano(f,B,g) : RngIntElt,GRBskt -> 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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