- First Examples
- 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 (H121E1)
- Algebraic Curves
- 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
- Basic Attributes
- Basic Invariants
- 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 (H121E4)
- Ordinary Plane Curves
- Local Geometry
- Creation of Points on Curves
- Operations at a Point
- Singularity Analysis
- Resolution of Singularities
- Log Canonical Thresholds
- 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 (H121E10)
- Example Crv_crv:int-nmbrs (H121E11)
- Global Geometry
- Maps and Curves
- Automorphism Groups of Curves
- Function Fields
- Function Fields
- FunctionField(C) : Crv -> FldFunFracSch
- Curve(F) : FldFunFracSch -> Crv
- F ! r : FldFunFracSch, RngElt -> FldFunFracSchElt
- ProjectiveFunction(f) : FldFunFracSchElt -> RngFunFracElt
- Example Crv_ff-creation-example (H121E24)
- 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 (H121E25)
- 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
- Zeta Functions of Curves
- Representations of the Function Field
- Differentials
- Creation of Differentials
- 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 (H121E27)
- Divisors
- Places
- Sets of Places
- 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 (H121E28)
- Zeros(f) : FldFunFracSchElt[Crv] -> SeqEnum[PlcCrvElt]
- Zeros(C, f) : Crv, RngElt -> [PlcCrvElt]
- CommonZeros(L) : [FldFunFracSchElt[Crv]] -> [PlcCrvElt]
- Example Crv_zeros-and-poles (H121E29)
- 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
- 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 (H121E30)
- 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 (H121E31)
- 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 (H121E32)
- D eq E : DivCrvElt, DivCrvElt -> BoolElt
- AreLinearlyEquivalent(D,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 (H121E33)
- Other Operations on Divisors
- Linear Equivalence of Divisors
- Linear Equivalence and Class Group
- Riemann--Roch Spaces
- 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 (H121E37)
- Advanced Examples
- Curves over Global Fields
- Minimal Degree Functions and Plane Models
- Bibliography
V2.28, 13 July 2023