Rational Curves and Conics

• Introduction
• Rational Curves and Conics
  • Rational Curve and Conic Creation
    • Conic(coeffs): [RngElt] → CrvCon
    • Conic(M): Mtrx → CrvCon
    • Conic(X, f): Prj, RngMPolElt → CrvCon
    • IsConic(S): Sch → BoolElt, CrvCon
    • RationalCurve(X, f): Prj, RngMPolElt → CrvRat
    • IsRationalCurve(S): Sch → BoolElt, CrvRat
    • Example: Creating a Conic from a Curve
  • Access Functions
    • DefiningPolynomial(C): Crv → RngMPolElt
    • DefiningIdeal(C): Crv → RngMPol
    • BaseRing(C): Crv → Rng
    • BaseField(C): Crv → Rng
    • Category(C): Crv → Cat
    • Type(C): Crv → Cat
  • Rational Curve and Conic Examples
    • Example: Singular Curve of Genus Zero
    • Example: Testing a Curve for Rationality
    • Example: Diagonal Equations for Conics
• Conics
  • Elementary Invariants
    • Discriminant(C): CrvCon → FldElt
  • Alternative Defining Polynomials
    • LegendrePolynomial(C): CrvCon → RngMPolElt, ModMatRngElt
    • ReducedLegendrePolynomial(C): CrvCon → RngMPolElt, ModMatRngElt
  • Alternative Models
    • LegendreModel(C): CrvCon → CrvCon, MapIsoSch
    • ReducedLegendreModel(C): CrvCon → CrvCon, MapIsoSch
  • Other Functions on Conics
    • MinimalModel(C): CrvCon → CrvCon, Map
    • Example: Minimal Model of a Conic
• Local-Global Correspondence
  • Local Conditions for Conics
    • BadPrimes(C): CrvCon → SeqEnum
  • Local Solubility
    • IsLocallySolvable(C): CrvCon → BoolElt
  • Norm Residue Symbol
    • NormResidueSymbol(a, b, p): FldRatElt, FldRatElt, RngIntElt → RngIntElt
    • NormResidueSymbol(a, b, p): RngIntElt, RngIntElt, RngIntElt → RngIntElt
    • HilbertSymbol(a, b, p : parameters): FldRatElt, FldRatElt, RngIntElt → RngIntElt
    • HilbertSymbol(a, b, p : parameters): RngIntElt, RngIntElt, RngIntElt → RngIntElt
    • HilbertSymbol(a, b, p): FldAlgElt, FldAlgElt, RngOrdIdl → RngIntElt
    • Example: Bad Primes via Norm Residue Symbols
• Rational Points on Conics
  • Finding Points
    • HasRationalPoint(C): CrvCon → BoolElt, Pt
    • RationalPoint(C): CrvCon → Pt
    • Random(C : parameters): CrvCon → Pt
    • Points(C : parameters): CrvCon → SetIndx
    • RationalPoints(C : parameters): CrvCon → SetIndx
    • Example: Points on Three Legendre Conics
  • Point Reduction
    • IsReduced(p): Pt → BoolElt
    • Reduction(p): Pt → Pt
    • Example: Reduced Points on a Conic
    • Example: Finding Points on a Conic
• Isomorphisms
  • Isomorphisms with Standard Models
    • Conic(C): Crv → MapSch
    • Example: Conic Model of a Rational Curve
    • ParametrizationMatrix(C): CrvCon → ModMatRngElt
    • Example: Parametrisation Matrix of a Conic
    • Parametrization(C): CrvCon → MapSch
    • Parametrization(C, P): CrvCon, Crv → MapSch
    • Parametrization(C, p): Crv, Pt → MapSch
    • Parametrization(C, p): Crv, PlcCrvElt → MapSch
    • Parametrization(C, p, P): CrvCon, Pt, Crv → MapSch
    • Parametrization(C, p, P): CrvRat, Pt, Crv → MapSch
    • Parametrization(C, p, P): CrvCon, PlcCrvElt, Crv → MapSch
    • Parametrization(C, p, P): CrvRat, PlcCrvElt, Crv → MapSch
    • ParametrizeOrdinaryCurve(C): Crv → MapSch
    • ParametrizeOrdinaryCurve(C, p): Crv, Pt → MapSch
    • ParametrizeOrdinaryCurve(C, p): Crv, PlcCrvElt → MapSch
    • ParametrizeOrdinaryCurve(C, p, I): Crv, Pt, RngMPol → MapSch
    • ParametrizeOrdinaryCurve(C, p, I): Crv, PlcCrvElt, RngMPol → MapSch
    • ParametrizeRationalNormalCurve(C): Crv → MapSch
    • Example: Parametrising a Rational Curve
• Automorphisms
  • Automorphisms of Rational Curves
    • Automorphism(C, S, T): CrvRat, SetIndx, SetIndx → MapIsoSch
  • Automorphisms of Conics
    • QuaternionAlgebra(C): CrvCon → AlgQuat
    • Automorphism(C, a): CrvCon, AlgQuatElt → MapIsoSch
    • Example: Automorphisms of a Conic Without Points
• Bibliography