Ambient Spaces
AffineSpace(k,n) : Rng, RngIntElt -> Aff
ProjectiveSpace(k,n) : Rng,RngIntElt -> Prj
DirectProduct(A,B) : Prj,Prj -> PrjProd,SeqEnum
RuledSurface(k,n) : Rng,RngIntElt -> PrjScrl
CoordinateRing(A) : Sch -> RngMPol
FunctionField(A) : Aff -> FldFunFracSch
A ! [a,...] : Sch,[RngElt] -> Pt
Origin(A) : Aff -> Pt
Coordinates(p) : Pt -> SeqEnum
Example Crv_plane-points (H114E1)
Creation
Curve(A,f) : Sch, RngMPolElt -> CrvPln
Curve(A,I) : Sch, RngMPol -> Crv
Curve(X,S) : Sch, SeqEnum -> Crv
IsCurve(X) : Sch -> BoolElt,Crv
Curve(X) : Sch -> Crv
Line(C,p,q) : CrvPln, Pt,Pt -> CrvPln
Conic(P,S) : Prj, {Pt} -> Crv
Union(C,D) : Sch,Sch -> Sch
Base Change
BaseChange(C, K) : Sch,Rng -> Sch
BaseChange(C, m) : Sch,Map -> Sch
BaseChange(C, A) : Sch,Sch -> Sch
BaseChange(C, n) : Sch, RngIntElt -> Sch
Example Crv_curve-base-change (H114E2)
Basic Attributes
AmbientSpace(C) : Sch -> Sch
BaseRing(C) : Sch -> Rng
DefiningPolynomial(C) : Sch -> RngMPolElt
DefiningIdeal(C) : Sch -> RngMPol
CoordinateRing(C) : Sch -> Rng
Degree(C) : Sch -> RngIntElt
JacobianIdeal(C) : Sch -> RngMPol
JacobianMatrix(C) : Sch -> ModMatRngElt
HessianMatrix(C) : Sch -> Mtrx
Example Crv_curve-hessian (H114E3)
Basic Invariants
IsReduced(C) : Sch -> BoolElt
IsIrreducible(C) : Sch -> BoolElt
IsSingular(C) : Sch -> BoolElt
IsNonsingular(C) : Sch -> BoolElt
Random Curves
RandomNodalCurve(d, g, P) : RngIntElt, RngIntElt, Prj -> CrvPln
IsNodalCurve(C) : Crv-> BoolElt
RandomOrdinaryPlaneCurve(d, S, P) : RngIntElt, SeqEnum, Prj -> CrvPln, RngMPol
RandomCurveByGenus(g, K) : RngIntElt, Fld -> Crv
Example Crv_random-curves (H114E4)
Ordinary Plane Curves
HasOnlyOrdinarySingularities(C) : CrvPln -> BoolElt, RngIntElt, RngMPol
HasOnlyOrdinarySingularitiesMonteCarlo(C) : CrvPln -> BoolElt, RngIntElt
AdjointIdeal(C) : Crv -> RngMPol
AdjointIdealForNodalCurve(C) : Crv -> RngMPol
AdjointLinearSystemFromIdeal(I, d) : RngMPol, RngIntElt -> LinearSys
CanonicalLinearSystemFromIdeal(I, d) : RngMPol, RngIntElt -> LinearSys
CanonicalLinearSystem(C) : Crv -> LinearSys
Example Crv_ordinary-curves (H114E5)
Creation of Points on Curves
C ! [a,...] : Crv,[RngElt] -> Pt
C(L) ! [a,...] : SetPt,[RngElt] -> Pt
Curve(p) : Pt -> Crv
Curve(P) : SetPt -> Crv
Coordinates(p) : Pt -> SeqEnum
p[i] : Pt, RngIntElt -> RngElt
p eq q : Pt,Pt -> BoolElt
FormalPoint(P) : Pt -> Pt
Operations at a Point
p in C : Pt,Sch -> BoolElt
IsNonsingular(p) : Sch,Pt -> BoolElt
IsSingular(p) : Sch,Pt -> BoolElt
IsInflectionPoint(p) : Sch,Pt -> BoolElt,RngIntElt
TangentLine(p) : Pt -> Crv
TangentCone(p) : Pt -> Sch
IsTangent(C,D,p) : Sch,Sch,Pt -> BoolElt
Singularity Analysis
Multiplicity(p) : Sch,Pt -> RngIntElt
IsDoublePoint(p) : Pt -> BoolElt
IsOrdinarySingularity(p) : Sch,Pt -> BoolElt
IsNode(p) : Pt -> BoolElt
IsCusp(p) : Crv,Pt -> BoolElt
IsAnalyticallyIrreducible(p) : CrvPln,Pt -> BoolElt
Example Crv_curve-iscusp (H114E6)
Resolution of Singularities
Blowup(C) : CrvPln -> CrvPln, CrvPln
Blowup(C,M) : CrvPln,Mtrx -> CrvPln, RngIntElt, RngIntElt
Example Crv_weighted-blowup (H114E7)
Log Canonical Thresholds
LogCanonicalThreshold(C) : Sch -> FldRatElt, BoolElt
LogCanonicalThresholdAtOrigin(C) : Sch -> FldRatElt
LogCanonicalThreshold(C, P) : Sch, Pt -> FldRatElt
LogCanonicalThresholdOverExtension(C) : Sch -> FldRatElt
Example Crv_lct-projective-plane (H114E8)
Example Crv_lct-over-ext (H114E9)
Local Intersection Theory
IsIntersection(C,D,p) : Sch,Sch,Pt -> BoolElt
IsTransverse(C,D,p) : Sch,Sch,Pt -> BoolElt
IntersectionNumber(C,D,p) : Sch,Sch,Pt -> RngIntElt
IntersectionNumbers(C,D) : CrvPln,CrvPln -> List
Example Crv_local-intersection-example (H114E10)
Example Crv_crv:int-nmbrs (H114E11)
Genus and Singularities
Genus(C) : Crv -> RngIntElt
ArithmeticGenus(C) : Crv -> RngIntElt
NumberOfPunctures(C): CrvPln -> RngIntElt
SingularPoints(C) : Sch -> SetIndx
HasSingularPointsOverExtension(C) : Sch -> BoolElt
Flexes(C) : Sch -> Sch
C eq D : Sch,Sch -> BoolElt
IsSubscheme(C,D) : Sch,Sch -> BoolElt
Example Crv_crv-genus (H114E12)
Projective Closure and Affine Patches
ProjectiveClosure(A): Sch -> Sch
ProjectiveClosure(C) : Sch -> Sch
Example Crv_proj-cl-commutes (H114E13)
LineAtInfinity(A) : Aff -> CrvPln
PointsAtInfinity(C) : Crv -> SetEnum
AffinePatch(C,i) : Crv,RngIntElt -> SeqEnum
Example Crv_second-affine-patch (H114E14)
Special Forms of Curves
IsEllipticWeierstrass(C) : Crv -> BoolElt
IsHyperellipticWeierstrass(C) : Crv -> BoolElt
EllipticCurve(C) : Crv -> CrvEll, MapSch
IsHyperelliptic(C) : Crv -> BoolElt, CrvHyp, MapSch
Example Crv_is_hyperelliptic (H114E15)
Elementary Maps
IdentityAutomorphism(A) : Sch -> AutSch
TranslationToInfinity(C,p) : Crv,Pt -> Crv,AutSch
Example Crv_translation-to-infinity (H114E16)
EvaluateByPowerSeries(m, P) : MapSch, Pt -> Pt
Example Crv_maps-point_pow_eval (H114E17)
Maps Induced by Morphisms
Degree(m) : MapSch -> RngIntElt
RamificationDivisor(m) : MapSch -> DivCrvElt
Pullback(phi, X) : MapSch, FldFunFracSchElt -> FldFunFracSchElt
Pushforward(phi, X) : MapSch, FldFunFracSchElt -> FldFunFracSchElt
Example Crv_map-push-pull (H114E18)
Group Creation Functions
AutomorphismGroup(C) : Crv -> GrpAutCrv
AutomorphismGroup(C,auts) : Crv, SeqEnum -> GrpAutCrv
Automorphisms(C) : Crv -> SeqEnum
IsIsomorphic(C, D) : Crv, Crv -> BoolElt,MapSch
Isomorphisms(C, D) : Crv, Crv -> SeqEnum
Automorphisms
A . i : GrpAutCrv, RngIntElt -> GrpAutCrvElt
Identity(A) : GrpAutCrv -> GrpAutCrvElt
A ! f : GrpAutCrv, MapSch -> GrpAutCrvElt
Order(f) : GrpAutCrvElt -> RngIntElt
Inverse(f) : GrpAutCrvElt -> GrpAutCrvElt
f * g : GrpAutCrvElt, GrpAutCrvElt -> GrpAutCrvElt
f ^ n : GrpAutCrvElt, RngIntElt -> GrpAutCrvElt
g eq h : GrpAutoElt, GrpAutoElt -> BoolElt
g ne h : GrpAutoElt, GrpAutoElt -> BoolElt
SchemeMap(f) : GrpAutCrvElt -> MapAutSch
Automorphism Group Operations
Curve(A) : GrpAutCrv -> Crv
Order(A) : GrpAutCrv -> RngIntElt
FactoredOrder(A) : GrpAutCrv -> [ <RngIntElt, RngIntElt> ]
NumberOfGenerators(A) : GrpAutCrv -> RngIntElt
Generators(A) : GrpAutCrv -> SeqEnum
PermutationGroup(A) : GrpAutCrv -> GrpPerm
PermutationRepresentation(A) : GrpAutCrv -> GrpPerm, Map
MatrixRepresentation(A) : GrpAutCrv -> Grpmat, Map, SeqEnum
a in A: GrpAutCrvElt, GrpAutCrv -> BoolElt
A subset B: GrpAutCrv, GrpAutCrv -> BoolElt
Pullbacks and Pushforwards
f(X): GrpAutCrvElt, Pt -> Pt
X @@ f: FldFunFracSchElt, GrpAutCrvElt -> FldFunFracSchElt
Example Crv_crv_autos (H114E19)
Example Crv_crv-iso (H114E20)
Example Crv_crv-iso (H114E21)
Quotients of Curves
CurveQuotient(G): GrpAutCrv -> Crv, MapSch
Example Crv_crv_quots (H114E22)
Example Crv_crv_quots (H114E23)
Function Fields
FunctionField(C) : Crv -> FldFunFracSch
Curve(F) : FldFunFracSch -> Crv
F ! r : FldFunFracSch, RngElt -> FldFunFracSchElt
ProjectiveFunction(f) : FldFunFracSchElt -> RngFunFracElt
Example Crv_ff-creation-example (H114E24)
p @ f : Pt, FldFunFracSchElt -> RngElt
Expand(f, p) : FldFunFracSchElt[Crv], PlcCrvElt -> RngSerElt, FldFunFracSchElt
Completion(F, p) : FldFunFracSch[Crv], PlcCrvElt -> RngSer, Map
Degree(f) : FldFunFracSchElt[Crv] -> RngIntElt
Valuation(f, p) : RngElt, Pt -> RngIntElt
Valuation(p) : Pt -> Map
UniformizingParameter(p) : Pt -> FldFunFracSchElt
Module(S) : [FldFunFracSchElt[Crv]] -> Mod, Map, [ModElt]
Relations(S) : [FldFunFracSchElt[Crv]] -> ModTupRng
Genus(C) : Crv -> RngIntElt
FieldOfGeometricIrreducibility(C) : Crv -> Rng, Map
IsAbsolutelyIrreducible(C) : Crv -> BoolElt
DimensionOfFieldOfGeometricIrreducibility(C): Crv -> RngIntElt
Example Crv_ff-elements-example (H114E25)
GapNumbers(C) : Crv -> [RngIntElt]
WronskianOrders(C) : Crv -> [RngIntElt]
NumberOfPlacesOfDegreeOverExactConstantField(C, m) : Crv[FldFin], RngIntElt -> RngIntElt
NumberOfPlacesOfDegreeOneOverExactConstantField(C) : Crv[FldFin] -> RngIntElt
NumberOfPlacesOfDegreeOneOverExactConstantField(C, m) : Crv[FldFin], RngIntElt -> RngIntElt
NumberOfPlacesOfDegreeOneECFBound(C) : Crv -> RngIntElt
DivisorOfDegreeOne(C) : Crv[FldFin] -> DivCrvElt
SerreBound(C) : Crv[FldFin] -> RngIntElt
LPolynomial(C) : Crv[FldFin] -> RngUPolElt
Representations of the Function Field
AlgorithmicFunctionField(F) : FldFunFracSch -> FldFun, Map
FunctionFieldPlace(p) : PlcCrvElt -> PlcFunElt
Creation of Differentials
DifferentialSpace(C) : Crv -> DiffCrv
SpaceOfDifferentialsFirstKind(C) : Crv -> ModFld, Map
BasisOfDifferentialsFirstKind(C) : Crv -> [DiffCrvElt]
DifferentialSpace(D) : DivCrvElt -> ModFld,Map
DifferentialBasis(D) : DivCrvElt -> [DiffCrvElt]
Differential(a) : FldFunFracSchElt -> DiffCrvElt
Operations on Differentials
Identity(S) : DiffCrv -> DiffCrvElt
Curve(S) : DiffCrv -> Crv
Curve(a) : DiffCrvElt -> Crv
S eq T : DiffCrv,DiffCrv -> BoolElt
a eq b : DiffCrvElt,DiffCrvElt -> BoolElt
a in S : Any,DiffCrv -> BoolElt
IsExact(a) : DiffCrvElt -> BoolElt
IsZero(a) : DiffCrvElt -> BoolElt
Valuation(d, P) : DiffCrvElt, PlcCrvElt -> RngIntElt
Residue(d, P): DiffCrvElt, PlcCrvElt -> RngElt
Divisor(d) : DiffCrvElt -> DivCrvElt
Module(L) : [DiffCrvElt] -> Mod, Map, [ ModElt ]
Relations(L) : [DiffCrvElt] -> ModTupFld
Cartier(a) : DiffCrvElt -> DiffCrvElt
CartierRepresentation(C) : Crv -> AlgMatElt, SeqEnum[DiffCrvElt]
Example Crv_curve-differentials (H114E26)
Sets of Places
Places(C) : Crv -> PlcCrv
Curve(P) : PlcCrv -> Crv
P eq Q : PlcCrv, PlcCrv -> BoolElt
Places
Places(C, m) : Crv[FldFin], RngIntElt -> SeqEnum
HasPlace(C, m) : Crv[FldFin], RngIntElt -> BoolElt,PlcCrvElt
Place(p) : Pt -> PlcCrvElt
Places(p) : Pt -> SeqEnum
Place(C, I) : Crv, RngMPol -> PlcCrvElt
WeierstrassPlaces(C) : Crv -> [PlcCrvElt]
Place(Q) : [FldFunFracSchElt] -> PlcCrvElt
Ideal(P) : PlcCrvElt -> RngMPol
TwoGenerators(P) : PlcCrvElt -> FldFunFracSchElt, FldFunFracSchElt
Example Crv_place-equations (H114E27)
Zeros(f) : FldFunFracSchElt[Crv] -> SeqEnum[PlcCrvElt]
Zeros(C, f) : Crv, RngElt -> [PlcCrvElt]
CommonZeros(L) : [FldFunFracSchElt[Crv]] -> [PlcCrvElt]
Example Crv_zeros-and-poles (H114E28)
Curve(P) : PlcCrvElt -> Crv
RepresentativePoint(P) : PlcCrv -> Pt
P eq Q : PlcCrvElt, PlcCrvElt -> BoolElt
Valuation(f, P) : RngElt, PlcCrvElt -> RngIntElt
Valuation(P) : PlcCrvElt -> Map
Valuation(a, P) : DiffCrvElt, PlcCrvElt -> RngIntElt
Residue(a, P) : DiffCrvElt, PlcCrvElt -> RngElt
UniformizingParameter(P) : PlcCrvElt -> FldFunFracSchElt
IsWeierstrassPlace(P) : PlcCrvElt -> BoolElt
ResidueClassField(P) : PlcCrvElt -> Rng
Evaluate(a, P) : FldFunFracSchElt, PlcCrvElt -> RngElt
Lift(a, P) : RngElt, PlcCrvElt -> FldFunFracSchElt
Degree(P) : PlcCrvElt -> RngIntElt
GapNumbers(C, P) : Crv, PlcCrvElt -> [RngIntElt]
Parametrization(C, p) : Crv, Pt -> MapSch
Divisor Group
DivisorGroup(C) : Crv -> DivCrv
Curve(Div) : DivCrv -> Crv
Div1 eq Div2 : DivCrv, DivCrv -> BoolElt
Creation of Divisors
DivisorGroup(D) : DivCrvElt -> DivCrv
Curve(D) : DivCrvElt -> Crv
Identity(D) : DivCrv -> DivCrvElt
Div ! p : DivCrv, PlcCrvElt -> DivCrvElt
Divisor(D, S) : DivCrv, SeqEnum -> DivCrvElt
Example Crv_divisor-equations (H114E29)
PrincipalDivisor(C, f) : Crv, RngElt -> DivCrvElt
Divisor(a) : DiffCrvElt -> DivCrvElt
Divisor(C, X) : Crv, Sch -> DivCrvElt
Divisor(C, p, q) : Crv,Pt,Pt -> DivCrvElt
Divisor(C, I) : Crv, RngMPol -> DivCrvElt
Decomposition(D) : DivCrvElt -> SeqEnum
Support(D) : DivCrvElt -> SeqEnum, SeqEnum
Example Crv_divisor1 (H114E30)
CanonicalDivisor(C) : Crv -> DivCrvElt
RamificationDivisor(C) : Crv -> DivCrvElt
Arithmetic of Divisors
Quotrem(D, n) : DivCrvElt, RngIntElt -> DivCrvElt, DivCrvElt
Degree(D) : DivCrvElt -> RngIntElt
IsEffective(D) : DivCrvElt -> BoolElt
Numerator(D) : DivCrvElt -> DivCrvElt
SignDecomposition(D) : DivCrvElt -> DivElt,DivElt
Example Crv_divisor2 (H114E31)
D eq E : DivCrvElt, DivCrvElt -> BoolElt
IsZero(D) : DivCrvElt -> BoolElt
IsCanonical(D) : DivCrvElt -> BoolElt, DiffCrvElt
GCD(D1, D2) : DivCrvElt, DivCrvElt -> DivCrvElt
LCM(D1, D2) : DivCrvElt, DivCrvElt -> DivCrvElt
Example Crv_canonical_divisor (H114E32)
Other Operations on Divisors
Ideal(D) : DivCrvElt -> RngMPol
Valuation(D,p) : DivCrvElt, Pt -> DivCrvElt
ComplementaryDivisor(D,p) : DivCrvElt,Pt -> DivCrvElt
Linear Equivalence of Divisors
Linear Equivalence and Class Group
IsPrincipal(D) : DivCrvElt -> BoolElt, FldFunFracSchElt
IsLinearlyEquivalent(D1,D2) : DivCrvElt,DivCrvElt -> BoolElt
IsHypersurfaceDivisor(D) : DivCrvElt -> BoolElt, RngElt, RngIntElt
Example Crv_is-hyper-surfacr-divisor-example (H114E33)
ClassGroup(C) : Crv[FldFin] -> GrpAb, Map, Map
ClassNumber(C) : Crv[FldFin] -> RngIntElt
GlobalUnitGroup(C) : Crv[FldFin] -> GrpAb, Map
Example Crv_divisor-class-group-example (H114E34)
ClassGroupAbelianInvariants(C) : Crv[FldFin] -> [RngIntElt]
ClassGroupPRank(C) : Crv[FldFin] -> RngIntElt
HasseWittInvariant(C) : Crv[FldFin] -> RngIntElt
Riemann--Roch Spaces
Reduction(D) : DivCrvElt -> DivCrvElt, RngIntElt, DivCrvElt, FldFunFracSchElt
RiemannRochSpace(D) : DivCrvElt -> ModFld,Map
Basis(D) : DivCrvElt -> SeqEnum
ShortBasis(D) : DivCrvElt -> SeqEnum
Dimension(D) : DivCrvElt -> RngIntElt
DifferentialSpace(D) : DivCrvElt -> ModFld, Map
DifferentialBasis(D) : DivCrvElt -> SeqEnum
IndexOfSpeciality(D) : DivCrvElt -> RngIntElt
IsSpecial(D) : DivCrvElt -> BoolElt
GapNumbers(D) : DivCrvElt -> SeqEnum
GapNumbers(p) : Pt -> SeqEnum
WeierstrassPlaces(D) : DivCrvElt -> SeqEnum
WronskianOrders(D) : DivCrvElt -> SeqEnum
RamificationDivisor(D) : DivCrvElt -> DivCrvElt
DivisorMap(D) : DivCrvElt -> MapSch
CanonicalMap(C) : Crv -> MapSch
CanonicalImage(C, phi) : Crv, MapSch -> Crv, BoolElt
Example Crv_canonical-map (H114E35)
Index Calculus
IndexCalculus(D1, D2, D0, np) : DivCrvElt, DivCrvElt, DivCrvElt, RngIntElt -> RngIntElt
IndexCalculusMatrix(D1, D2, D0, n, rr) : DivCrvElt, DivCrvElt, DivCrvElt, RngIntElt, RngIntElt -> MtrxSprs, SeqEnum, SeqEnum, DivCrvElt, DivCrvElt, RngIntElt, RngIntElt
MultiplyDivisor(n, D , D0) : RngIntElt, DivCrvElt, DivCrvElt -> DivCrvElt
Example Crv_indexcalculus (H114E36)
Trigonal Curves
Example Crv_trigonal-curve (H114E37)
Algebraic Geometric Codes
Example Crv_klein-quartic-code (H114E38)
Finding Rational Points
PointsCubicModel(C, B : parameters) : Crv, RngIntElt -> SeqEnum
Example Crv_points-cubic-model (H114E39)
Regular Models of Arithmetic Surfaces
Creation of Regular Models
RegularModel(C, P) : Crv, Any -> CrvRegModel
Using Regular Models
IntersectionMatrix(M) : CrvRegModel -> Mtrx, SeqEnum
ComponentGroup(M) : CrvRegModel -> GrpAb
PointOnRegularModel(M, x) : CrvRegModel, Pt -> SeqEnum, SeqEnum, Tup
Minimization and Reduction
ReduceCluster(X) : SeqEnum -> SeqEnum, Mtrx, Mtrx
ReducePlaneCurve(f) : MPolElt -> RngMPolElt, Mtrx
Minimization and Reduction for Plane Quartics
MinimizePlaneQuartic(f,p) : MPolElt, RngIntElt -> RngMPolElt, Mtrx
MinimizeReducePlaneQuartic(f) : MPolElt -> RngMPolElt, Mtrx
Example Crv_minredplanequartic (H114E40)
Minimal Degree Functions and Plane Models
General Functions and Clifford Index One
GenusAndCanonicalMap(C) : Crv -> RngIntElt, BoolElt, MapSch
CliffordIndexOne(C) : Crv -> MapSch
Example Crv_gon-gen-ex (H114E41)
Small Genus Functions
Genus2GonalMap(C) : Crv -> MapSch
Genus3GonalMap(C) : Crv -> RngIntElt, MapSch
Genus4GonalMap(C) : Crv -> RngIntElt, MapSch
Genus5GonalMap(C) : Crv -> RngIntElt, MapSch, Crv, UserProgram
Genus6GonalMap(C) : Crv -> RngIntElt, RngIntElt, MapSch, MapSch
Example Crv_gon-sm-gen-ex (H114E42)
Small Genus Plane Models
Genus6PlaneCurveModel(C) : Crv -> BoolElt, MapSch
Genus5PlaneCurveModel(C) : Crv -> BoolElt, MapSch
Example Crv_gon-pln_mod-ex (H114E43)
Bibliography
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Wed Apr 24 15:09:57 EST 2013