Projective Plane Curves
ProjectivePlaneCurve — TypeProjectivePlaneCurve <: AbsProjectiveCurveA reduced curve in the projective plane.
Examples
julia> R, (x,y,z) = graded_polynomial_ring(QQ, [:x, :y, :z]);
julia> C = plane_curve(y^3*x^6 - y^6*x^2*z)
Projective plane curve
defined by 0 = x^5*y - x*y^4*z
Projective plane curves are modeled in Oscar as projective algebraic sets. See AbsProjectiveAlgebraicSet(@ref). In addition to the methods for algebraic sets and curves the following methods special to plane curves are available.
defining_equation — Methoddefining_equation(C::AffinePlaneCurve)Return the defining equation of C.
defining_equation(C::ProjectivePlaneCurve)Return the defining equation of the (reduced) plane curve C.
degree — Methoddegree(C::ProjectivePlaneCurve)Return the degree of the defining polynomial of C.
common_components — Methodcommon_components(C::S, D::S) where {S<:ProjectivePlaneCurve}Return the projective plane curve consisting of the common components of C and D, or an empty vector if they do not have a common component.
multiplicity — Methodmultiplicity(C::ProjectivePlaneCurve{S}, P::AbsProjectiveRationalPoint)Return the multiplicity of C at P.
tangent_lines — Methodtangent_lines(C::ProjectivePlaneCurve{S}, P::AbsProjectiveRationalPoint) where S <: FieldElemReturn the tangent lines at P to C with their multiplicity.
intersection_multiplicity — Methodintersection_multiplicity(C::S, D::S, P::AbsProjectiveRationalPoint) where S <: ProjectivePlaneCurveReturn the intersection multiplicity of C and D at P.
is_transverse_intersection — Methodis_transverse_intersection(C::S, D::S, P::AbsProjectiveRationalPoint) where S <: ProjectivePlaneCurveReturn true if C and D intersect transversally at P and false otherwise.