Algebraic Curves
- First Examples
- Ambient Spaces
AffineSpace(k,n): Rng, RngIntElt → AffAffinePlane(k): Rng → AffProjectiveSpace(k,n): Rng, RngIntElt → PrjProjectivePlane(k): Rng → PrjDirectProduct(A,B): Prj, Prj → PrjProd, SeqEnumRuledSurface(k,n): Rng, RngIntElt → PrjScrlRuledSurface(k,a,b): Rng, RngIntElt, RngIntElt → PrjScrlCoordinateRing(A): Sch → RngMPolFunctionField(A): Aff → FldFunFracSchFunctionField(A): Prj → FldFunFracSchA ! [a,...]: Sch, [RngElt] → PtA(L) ! [a,...]: SetPt, [RngElt] → PtOrigin(A): Aff → PtCoordinates(p): Pt → SeqEnump[i]: Pt, RngIntElt → RngEltExample: Plane Points
- Algebraic Curves
- Creation
Curve(A,f): Sch, RngMPolElt → CrvPlnCurve(A,I): Sch, RngMPol → CrvCurve(X,S): Sch, SeqEnum → CrvIsCurve(X): Sch → BoolElt, CrvCurve(X): Sch → CrvLine(C,p,q): CrvPln, Pt, Pt → CrvPlnLine(P,S): Prj, \{Pt\} → SchConic(P,S): Prj, \{Pt\} → CrvUnion(C,D): Sch, Sch → Sch
- Base Change
BaseChange(C, K): Sch, Rng → SchBaseChange(C, m): Sch, Map → SchBaseChange(C, A): Sch, Sch → SchBaseChange(C, A, m): Sch, Sch, Map → SchBaseChange(C, n): Sch, RngIntElt → SchExample: Curve Base Change
- Basic Attributes
- Basic Invariants
- Random Curves
RandomNodalCurve(d, g, P): RngIntElt, RngIntElt, Prj → CrvPlnIsNodalCurve(C): Crv → BoolEltRandomOrdinaryPlaneCurve(d, S, P): RngIntElt, SeqEnum, Prj → CrvPln, RngMPolRandomCurveByGenus(g, K): RngIntElt, Fld → CrvExample: Random Curves
- Ordinary Plane Curves
HasOnlyOrdinarySingularities(C): CrvPln → BoolElt, RngIntElt, RngMPolHasOnlyOrdinarySingularitiesMonteCarlo(C): CrvPln → BoolElt, RngIntEltAdjointIdeal(C): Crv → RngMPolAdjointIdealForNodalCurve(C): Crv → RngMPolAdjointLinearSystemForNodalCurve(C, d): Crv, RngIntElt → LinearSysAdjointLinearSystemFromIdeal(I, d): RngMPol, RngIntElt → LinearSysCanonicalLinearSystemFromIdeal(I, d): RngMPol, RngIntElt → LinearSysCanonicalLinearSystem(C): Crv → LinearSysAdjointLinearSystem(C): Crv → LinearSysAdjoints(C,d): Crv, RngIntElt → LinearSysExample: Ordinary Curves
- Local Geometry
- Creation of Points on Curves
- Operations at a Point
p in C: Pt, Sch → BoolEltS in C: SeqEnum, Sch → BoolEltIsNonsingular(p): Pt → BoolEltIsNonsingular(C, p): Sch, Pt → BoolEltIsSingular(p): Pt → BoolEltIsSingular(C, p): Sch, Pt → BoolEltIsInflectionPoint(p): Pt → BoolElt, RngIntEltIsInflectionPoint(C, p): Sch, Pt → BoolElt, RngIntEltIsFlex(C, p): Sch, Pt → BoolElt, RngIntEltIsFlex(p): Pt → BoolElt, RngIntEltTangentLine(p): Pt → CrvTangentLine(C, p): Crv, Pt → CrvTangentCone(p): Pt → SchTangentCone(C, p): Sch, Pt → SchIsTangent(C, D, p): Sch, Sch, Pt → BoolElt
- Singularity Analysis
Multiplicity(p): Pt → RngIntEltMultiplicity(C, p): Sch, Pt → RngIntEltIsDoublePoint(p): Pt → BoolEltIsDoublePoint(C, p): Crv, Pt → BoolEltIsOrdinarySingularity(p): Pt → BoolEltIsOrdinarySingularity(C, p): Sch, Pt → BoolEltIsNode(p): Pt → BoolEltIsNode(C, p): Sch, Pt → BoolEltIsCusp(p): Pt → BoolEltIsCusp(C, p): Crv, Pt → BoolEltIsAnalyticallyIrreducible(p): Pt → BoolEltIsAnalyticallyIrreducible(C, p): CrvPln, Pt → BoolEltDeltaAdjustment(C, p): Sch, Pt → RngIntEltExample: Curve Iscusp
- Resolution of Singularities
- Log Canonical Thresholds
- Local Intersection Theory
IsIntersection(C,D,p): Sch, Sch, Pt → BoolEltIsTransverse(C,D,p): Sch, Sch, Pt → BoolEltIntersectionNumber(C,D,p): Sch, Sch, Pt → RngIntEltIntersectionNumbers(C,D): CrvPln, CrvPln → ListIntersectionNumbers(F,G): RngMPolElt, RngMPolElt → ListExample: Local Intersection ExampleExample: Crv:int Nmbrs
- Global Geometry
- Genus and Singularities
- Projective Closure and Affine Patches
- Special Forms of Curves
IsEllipticWeierstrass(C): Crv → BoolEltIsHyperellipticWeierstrass(C): Crv → BoolEltEllipticCurve(C): Crv → CrvEll, MapSchEllipticCurve(C,p): Crv, Pt → CrvEll, MapSchEllipticCurve(C,p): Crv, PlcCrvElt → CrvEll, MapSchIsHyperelliptic(C): Crv → BoolElt, CrvHyp, MapSchIsGeometricallyHyperelliptic(C): Crv → BoolElt, Crv, MapSchExample: Is Hyperelliptic
- Maps and Curves
- Elementary Maps
- Maps Induced by Morphisms
Degree(m): MapSch → RngIntEltRamificationDivisor(m): MapSch → DivCrvEltPullback(phi, X): MapSch, FldFunFracSchElt → FldFunFracSchEltPullback(phi, X): MapSch, DiffCrvElt → DiffCrvEltPullback(phi, X): MapSch, DivCrvElt → DivCrvEltPullback(phi, X): MapSch, PlcCrvElt → DivCrvEltPushforward(phi, X): MapSch, FldFunFracSchElt → FldFunFracSchEltPushforward(phi, X): MapSch, PlcCrvElt → DivCrvEltPushforward(phi, X): MapSch, DivCrvElt → DivCrvEltExample: Map Push Pull
- Automorphism Groups of Curves
- Group Creation Functions
- Automorphisms
- Automorphism Group Operations
- Pullbacks and Pushforwards
f(X): GrpAutCrvElt, Pt → Ptf(X): GrpAutCrvElt, FldFunFracSchElt → FldFunFracSchEltf(X): GrpAutCrvElt, PlcCrvElt → PlcCrvEltf(X): GrpAutCrvElt, DivCrvElt → DivCrvEltf(X): GrpAutCrvElt, DiffCrvElt → DiffCrvEltX @ f: Pt, GrpAutCrvElt → PtX @ f: FldFunFracSchElt, GrpAutCrvElt → FldFunFracSchEltX @ f: PlcCrvElt, GrpAutCrvElt → PlcCrvEltX @ f: DivCrvElt, GrpAutCrvElt → DivCrvEltX @ f: DiffCrvElt, GrpAutCrvElt → DiffCrvEltX @@ f: FldFunFracSchElt, GrpAutCrvElt → FldFunFracSchEltX @@ f: PlcCrvElt, GrpAutCrvElt → PlcCrvEltX @@ f: DivCrvElt, GrpAutCrvElt → DivCrvEltX @@ f: DiffCrvElt, GrpAutCrvElt → DiffCrvEltExample: Crv AutosExample: Crv IsoExample: Crv Iso
- Quotients of Curves
- Function Fields
- Function Fields
FunctionField(C): Crv → FldFunFracSchHasFunctionField(C): Crv → BoolEltCurve(F): FldFunFracSch → CrvF ! r: FldFunFracSch, RngElt → FldFunFracSchEltProjectiveFunction(f): FldFunFracSchElt → RngFunFracEltExample: Ff Creation Examplep @ f: Pt, FldFunFracSchElt → RngEltf(p): Pt, FldFunFracSchElt → RngEltEvaluate(f, p): RngElt, Pt → RngEltExpand(f, p): FldFunFracSchElt[Crv], PlcCrvElt → RngSerElt, FldFunFracSchEltCompletion(F, p): FldFunFracSch[Crv], PlcCrvElt → RngSer, MapDegree(f): FldFunFracSchElt[Crv] → RngIntEltValuation(f, p): RngElt, Pt → RngIntEltValuation(p): Pt → MapUniformizingParameter(p): Pt → FldFunFracSchEltModule(S): [FldFunFracSchElt[Crv]] → Mod, Map, [ModElt]Relations(S): [FldFunFracSchElt[Crv]] → ModTupRngRelations(S, m): [FldFunFracSchElt[Crv]], RngIntElt → ModTupRngGenus(C): Crv → RngIntEltFieldOfGeometricIrreducibility(C): Crv → Rng, MapIsAbsolutelyIrreducible(C): Crv → BoolEltDimensionOfFieldOfGeometricIrreducibility(C): Crv → RngIntEltExample: Ff Elements ExampleGapNumbers(C): Crv → [RngIntElt]WronskianOrders(C): Crv → [RngIntElt]NumberOfPlacesOfDegreeOverExactConstantField(C, m): Crv[FldFin], RngIntElt → RngIntEltNumberOfPlacesDegECF(C, m): Crv[FldFin], RngIntElt → RngIntEltNumberOfPlacesOfDegreeOneOverExactConstantField(C): Crv[FldFin] → RngIntEltNumberOfPlacesOfDegreeOneECF(C): Crv[FldFin] → RngIntEltNumberOfPlacesOfDegreeOneOverExactConstantField(C, m): Crv[FldFin], RngIntElt → RngIntEltNumberOfPlacesOfDegreeOneECF(C, m): Crv[FldFin], RngIntElt → RngIntEltNumberOfPlacesOfDegreeOneECFBound(C): Crv → RngIntEltNumberOfPlacesOfDegreeOneOverExactConstantFieldBound(C): Crv[FldFin] → RngIntEltNumberOfPlacesOfDegreeOneECFBound(C, m): Crv[FldFin], RngIntElt → RngIntEltNumberOfPlacesOfDegreeOneOverExactConstantFieldBound(C, m): Crv[FldFin], RngIntElt → RngIntEltDivisorOfDegreeOne(C): Crv[FldFin] → DivCrvEltSerreBound(C): Crv[FldFin] → RngIntEltSerreBound(C, m): Crv[FldFin], RngIntElt → RngIntEltIharaBound(C): Crv[FldFin] → RngIntEltIharaBound(C, m): Crv[FldFin], RngIntElt → RngIntElt
- Zeta Functions of Curves
- Representations of the Function Field
- Differentials
- Creation of Differentials
- Operations on Differentials
Identity(S): DiffCrv → DiffCrvEltCurve(S): DiffCrv → CrvCurve(a): DiffCrvElt → Crvf * x: RngElt, DiffCrvElt → DiffCrvEltx * f: DiffCrvElt, RngElt → DiffCrvEltx + y: DiffCrvElt, DiffCrvElt → DiffCrvElt- x: DiffCrvElt → DiffCrvEltx - y: DiffCrvElt, DiffCrvElt → DiffCrvEltx / r: DiffCrvElt, RngElt → DiffCrvEltx / y: DiffCrvElt, DiffCrvElt → FldFunFracSchEltS eq T: DiffCrv, DiffCrv → BoolElta eq b: DiffCrvElt, DiffCrvElt → BoolElta in S: Any, DiffCrv → BoolEltIsExact(a): DiffCrvElt → BoolEltIsZero(a): DiffCrvElt → BoolEltValuation(d, P): DiffCrvElt, PlcCrvElt → RngIntEltResidue(d, P): DiffCrvElt, PlcCrvElt → RngEltDivisor(d): DiffCrvElt → DivCrvEltModule(L): [DiffCrvElt] → Mod, Map, [ ModElt ]Relations(L): [DiffCrvElt] → ModTupFldRelations(L, m): [DiffCrvElt], RngIntElt → ModTupFldCartier(a): DiffCrvElt → DiffCrvEltCartier(a, r): DiffCrvElt, RngIntElt → DiffCrvEltCartierRepresentation(C): Crv → AlgMatElt, SeqEnum[DiffCrvElt]CartierRepresentation(C, r): Crv, RngIntElt → AlgMatElt, SeqEnum[DiffCrvElt]Example: Curve Differentials
- Divisors
- Places
- Sets of Places
- Places
Places(C, m): Crv[FldFin], RngIntElt → SeqEnumHasPlace(C, m): Crv[FldFin], RngIntElt → BoolElt, PlcCrvEltRandomPlace(C, m): Crv[FldFin], RngIntElt → BoolElt, PlcCrvEltPlace(p): Pt → PlcCrvEltPlaces(p): Pt → SeqEnumPlace(C, I): Crv, RngMPol → PlcCrvEltWeierstrassPlaces(C): Crv → [PlcCrvElt]Place(Q): [FldFunFracSchElt] → PlcCrvEltIdeal(P): PlcCrvElt → RngMPolTwoGenerators(P): PlcCrvElt → FldFunFracSchElt, FldFunFracSchEltExample: Place EquationsZeros(f): FldFunFracSchElt[Crv] → SeqEnum[PlcCrvElt]Poles(f): FldFunFracSchElt[Crv] → SeqEnum[PlcCrvElt]Zeros(C, f): Crv, RngElt → [PlcCrvElt]Poles(C, f): Crv, RngElt → [PlcCrvElt]CommonZeros(L): [FldFunFracSchElt[Crv]] → [PlcCrvElt]CommonZeros(C, L): Crv, [FldFunFracSchElt] → [PlcCrvElt]Example: Zeros And Polesp1 + p2: PlcCrvElt, PlcCrvElt → DivCrvElt- p1: PlcCrvElt → DivCrvEltp1 - p2: PlcCrvElt, PlcCrvElt → DivCrvEltk * p: RngIntElt, PlcCrvElt → DivCrvEltp div k: PlcCrvElt, RngIntElt → DivCrvEltp mod k: PlcCrvElt, RngIntElt → DivCrvEltQuotrem(p1, k): PlcCrvElt, RngIntElt → DivCrvElt, DivCrvEltCurve(P): PlcCrvElt → CrvRepresentativePoint(P): PlcCrv → PtP eq Q: PlcCrvElt, PlcCrvElt → BoolEltP ne Q: PlcCrvElt, PlcCrvElt → BoolEltP in S: Any, PlcCrv → BoolEltP notin S: Any, PlcCrv → BoolEltValuation(f, P): RngElt, PlcCrvElt → RngIntEltValuation(P): PlcCrvElt → MapValuation(a, P): DiffCrvElt, PlcCrvElt → RngIntEltResidue(a, P): DiffCrvElt, PlcCrvElt → RngEltUniformizingParameter(P): PlcCrvElt → FldFunFracSchEltIsWeierstrassPlace(P): PlcCrvElt → BoolEltIsWeierstrassPlace(D, P): DivCrvElt, PlcCrvElt → BoolEltResidueClassField(P): PlcCrvElt → RngEvaluate(a, P): FldFunFracSchElt, PlcCrvElt → RngEltLift(a, P): RngElt, PlcCrvElt → FldFunFracSchEltLift(i, P): Infty, PlcCrvElt → FldFunFracSchEltDegree(P): PlcCrvElt → RngIntEltGapNumbers(C, P): Crv, PlcCrvElt → [RngIntElt]GapNumbers(P): PlcCrvElt → [RngIntElt]Parametrization(C, p): Crv, Pt → MapSchParametrization(C, p): Crv, PlcCrvElt → MapSchParametrization(C, p, P): Crv, PlcCrvElt, Crv → MapSch
- Divisor Group
- Creation of Divisors
DivisorGroup(D): DivCrvElt → DivCrvCurve(D): DivCrvElt → CrvIdentity(D): DivCrv → DivCrvEltId(D): DivCrv → DivCrvEltD ! 0: DivCrv, RngIntElt → DivCrvEltDiv ! p: DivCrv, PlcCrvElt → DivCrvEltDiv ! p: DivCrv, Pt → DivCrvEltDivisor(p): PlcCrvElt → DivCrvEltDivisor(p): Pt → DivCrvEltDivisor(D, S): DivCrv, SeqEnum → DivCrvEltDivisor(C, S): Crv, SeqEnum → DivCrvEltDivisor(S): [<PlcCrvElt, RngIntElt>] → DivCrvEltExample: Divisor EquationsPrincipalDivisor(C, f): Crv, RngElt → DivCrvEltPrincipalDivisor(D, f): DivCrv, RngElt → DivCrvEltPrincipalDivisor(f): FldFunFracSchElt[Crv] → DivCrvEltDivisor(C, f): Crv, RngElt → DivCrvEltDivisor(D, f): DivCrv, RngElt → DivCrvEltDivisor(f): FldFunFracSchElt[Crv] → DivCrvEltDivisor(a): DiffCrvElt → DivCrvEltDivisor(C, X): Crv, Sch → DivCrvEltDivisor(D, X): DivCrv, Sch → DivCrvEltDivisor(C, p, q): Crv, Pt, Pt → DivCrvEltDivisor(D, p, q): DivCrv, Pt, Pt → DivCrvEltDivisor(C, I): Crv, RngMPol → DivCrvEltDivisor(D, I): DivCrv, RngMPol → DivCrvEltDecomposition(D): DivCrvElt → SeqEnumSupport(D): DivCrvElt → SeqEnum, SeqEnumExample: divisor1CanonicalDivisor(C): Crv → DivCrvEltRamificationDivisor(C): Crv → DivCrvElt
- Arithmetic of Divisors
D + E: DivCrvElt, DivCrvElt → DivCrvEltD + E: DivCrvElt, PlcCrvElt → DivCrvEltD + E: PlcCrvElt, DivCrvElt → DivCrvEltD + E: PlcCrvElt, PlcCrvElt → DivCrvElt- D: DivCrvElt → DivCrvElt- D: PlcCrvElt → DivCrvEltD - E: DivCrvElt, DivCrvElt → DivCrvEltD - E: DivCrvElt, PlcCrvElt → DivCrvEltD - E: PlcCrvElt, DivCrvElt → DivCrvEltD - E: PlcCrvElt, PlcCrvElt → DivCrvEltn * D: RngIntElt, DivCrvElt → DivCrvEltn * D: RngIntElt, PlcCrvElt → DivCrvEltD div n: DivCrvElt, RngIntElt → DivCrvEltD div n: PlcCrvElt, RngIntElt → DivCrvEltD mod n: DivCrvElt, RngIntElt → DivCrvEltD mod n: PlcCrvElt, RngIntElt → DivCrvEltQuotrem(D, n): DivCrvElt, RngIntElt → DivCrvElt, DivCrvEltDegree(D): DivCrvElt → RngIntEltIsEffective(D): DivCrvElt → BoolEltIsPositive(D): DivCrvElt → BoolEltNumerator(D): DivCrvElt → DivCrvEltDenominator(D): DivCrvElt → DivCrvEltSignDecomposition(D): DivCrvElt → DivElt, DivEltExample: divisor2d in D: Any, DivCrv → BoolEltd notin D: Any, DivCrv → BoolEltD eq E: DivCrvElt, DivCrvElt → BoolEltD ne E: DivCrvElt, DivCrvElt → BoolEltD lt E: DivCrvElt, DivCrvElt → BoolEltD le E: DivCrvElt, DivCrvElt → BoolEltD gt E: DivCrvElt, DivCrvElt → BoolEltD ge E: DivCrvElt, DivCrvElt → BoolEltAreLinearlyEquivalent(D,E): DivCrvElt, DivCrvElt → BoolEltIsZero(D): DivCrvElt → BoolEltIsCanonical(D): DivCrvElt → BoolElt, DiffCrvEltGCD(D1, D2): DivCrvElt, DivCrvElt → DivCrvEltGcd(D1, D2): DivCrvElt, DivCrvElt → DivCrvEltGreatestCommonDivisor(D1, D2): DivCrvElt, DivCrvElt → DivCrvEltLCM(D1, D2): DivCrvElt, DivCrvElt → DivCrvEltLcm(D1, D2): DivCrvElt, DivCrvElt → DivCrvEltLeastCommonMultiple(D1, D2): DivCrvElt, DivCrvElt → DivCrvEltExample: Canonical Divisor
- Other Operations on Divisors
- Linear Equivalence of Divisors
- Linear Equivalence and Class Group
- Riemann–Roch Spaces
Reduction(D): DivCrvElt → DivCrvElt, RngIntElt, DivCrvElt, FldFunFracSchEltReduction(D, A): DivCrvElt, DivCrvElt → DivCrvElt, RngIntElt, DivCrvElt, FldFunFracSchEltRiemannRochSpace(D): DivCrvElt → ModFld, MapBasis(D): DivCrvElt → SeqEnumShortBasis(D): DivCrvElt → SeqEnumDimension(D): DivCrvElt → RngIntEltDifferentialSpace(D): DivCrvElt → ModFld, MapDifferentialBasis(D): DivCrvElt → SeqEnumIndexOfSpeciality(D): DivCrvElt → RngIntEltIsSpecial(D): DivCrvElt → BoolEltGapNumbers(D): DivCrvElt → SeqEnumGapNumbers(D,p): DivCrvElt, PlcCrvElt → SeqEnumGapNumbers(p): Pt → SeqEnumWeierstrassPlaces(D): DivCrvElt → SeqEnumWeierstrassPoints(D): DivCrvElt → SeqEnumWronskianOrders(D): DivCrvElt → SeqEnumRamificationDivisor(D): DivCrvElt → DivCrvEltDivisorMap(D): DivCrvElt → MapSchDivisorMap(D,P): DivCrvElt, Prj → MapSchCanonicalMap(C): Crv → MapSchCanonicalMap(C,P): Crv, Prj → MapSchCanonicalImage(C, phi): Crv, MapSch → Crv, BoolEltCanonicalImage(C, eqns): Crv, SeqEnum → Crv, BoolEltExample: Canonical Map
- Index Calculus
IndexCalculus(D1, D2, D0, np): DivCrvElt, DivCrvElt, DivCrvElt, RngIntElt → RngIntEltIndexCalculus(D1, D2, D0, np, n, rr): DivCrvElt, DivCrvElt, DivCrvElt, RngIntElt, RngIntElt, RngIntElt → RngIntEltIndexCalculusMatrix({D1, D2, D0, }{n, rr}): DivCrvElt, DivCrvElt, DivCrvElt, RngIntElt, RngIntElt → MtrxSprs, SeqEnum, SeqEnum, DivCrvElt, DivCrvElt, RngIntElt, RngIntEltMultiplyDivisor(n, D , D0): RngIntElt, DivCrvElt, DivCrvElt → DivCrvEltExample: indexcalculus
- Advanced Examples
- Curves over Global Fields
- Minimal Degree Functions and Plane Models
