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