Plane curves (oscar-system#2988)

* plane curves as algebraic sets, remove exports for the old plane curves 

* Support radical membership for flattenings.

* Rework generation of charts.

* Add tests for projective schemes with empty charts.

* Remove deprecated call from ideal sheaves.

* Rework the homogenization and dehomogenization maps.


---------

Co-authored-by: Max Horn <max@quendi.de>
Co-authored-by: Matthias Zach <zach@math.uni-hannover.de>

docs/doc.main

@@ -201,7 +201,9 @@
         "AlgebraicGeometry/TropicalGeometry/curve.md",
      ],
      "Curves" => [
-         "AlgebraicGeometry/Curves/para_rational_curves.md",
+         "AlgebraicGeometry/Curves/AffinePlaneCurves.md",
+         "AlgebraicGeometry/Curves/ProjectiveCurves.md",
+         "AlgebraicGeometry/Curves/ProjectivePlaneCurves.md",
      ],
      "Surfaces" => [
         "AlgebraicGeometry/Surfaces/K3Surfaces.md",

docs/src/AlgebraicGeometry/Curves/AffinePlaneCurves.md

@@ -0,0 +1,16 @@
+```@meta
+CurrentModule = Oscar
+```
+
+# Affine plane curves
+
+```@docs
+AffinePlaneCurve
+defining_equation(C::AffinePlaneCurve{S, MPolyQuoRing{E}}) where {S, E}
+common_components(C::S, D::S) where {S<:AffinePlaneCurve}
+multiplicity(C::AffinePlaneCurve, P::AbsAffineRationalPoint)
+tangent_lines(C::AffinePlaneCurve, P::AbsAffineRationalPoint)
+intersection_multiplicity(C::AffinePlaneCurve, D::AffinePlaneCurve, P::AbsAffineRationalPoint)
+is_transverse_intersection(C::AffinePlaneCurve, D::AffinePlaneCurve, P::AbsAffineRationalPoint)
+projective_closure(C::AffinePlaneCurve)
+```

docs/src/AlgebraicGeometry/Curves/ProjectiveCurves.md

@@ -0,0 +1,10 @@
+```@meta
+CurrentModule = Oscar
+```
+
+# Projective Curves
+
+```@docs
+ProjectiveCurve
+invert_birational_map
+```

docs/src/AlgebraicGeometry/Curves/para_rational_curves.md → docs/src/AlgebraicGeometry/Curves/ProjectivePlaneCurves.md

@@ -1,6 +1,27 @@
 ```@meta
 CurrentModule = Oscar
 ```
+# Projective Plane Curves
+```@docs
+ProjectivePlaneCurve
+```
+
+Projective plane curves are modeled in Oscar as projective
+algebraic sets. See `AbsProjectiveAlgebraicSet`(@ref).
+In addition to the methods for algebraic sets
+the following methods special to plane curves are available.
+
+```@docs
+defining_equation(C::ProjectivePlaneCurve{S,MPolyQuoRing{T}}) where {S,T}
+degree(C::ProjectivePlaneCurve)
+common_components(C::S, D::S) where {S<:ProjectivePlaneCurve}
+multiplicity(C::ProjectivePlaneCurve, P::AbsProjectiveRationalPoint)
+tangent_lines(C::ProjectivePlaneCurve, P::AbsProjectiveRationalPoint)
+intersection_multiplicity(C::S, D::S, P::AbsProjectiveRationalPoint) where S <: ProjectivePlaneCurve
+is_transverse_intersection(C::S, D::S, P::AbsProjectiveRationalPoint) where S <: ProjectivePlaneCurve
+arithmetic_genus(C::ProjectivePlaneCurve)
+geometric_genus(C::AbsProjectiveCurve)
+```
 
 # Rational Parametrizations of Rational Plane Curves
 
@@ -48,36 +69,24 @@ curves only once. Its individual steps are interesting in their own right:
 
 See [Bhm99](@cite) and [BDLP17](@cite) for details and further references.
 
-## Creating Projective Plane Curves
 
-The data structures for algebraic curves in OSCAR are still under development
-and subject to change. Here is the current constructor for projective plane curves:
 
-```@docs
-ProjPlaneCurve(f::MPolyRingElem{T}) where {T <: FieldElem}
-```
-
-## The Genus of a Plane Curve
-
-```@docs
- geometric_genus(C::ProjectivePlaneCurve{T}) where T <: FieldElem
-```
 
 ## Adjoint Ideals of Plane Curves
 
 ```@docs
-adjoint_ideal(C::ProjPlaneCurve{QQFieldElem})
+adjoint_ideal(C::ProjectivePlaneCurve{QQField})
 ```
 
 ## Rational Points on Conics
 
 ```@docs
-rational_point_conic(D::ProjPlaneCurve{QQFieldElem})
+rational_point_conic(D::ProjectivePlaneCurve{QQField})
 ```
 ## Parametrizing Rational Plane Curves
 
 ```@docs
- parametrization_plane_curve(C::ProjPlaneCurve{QQFieldElem})
+parametrization(C::ProjectivePlaneCurve{QQField})
 ```
 
 

docs/src/AlgebraicGeometry/Schemes/ProjectiveSchemes.md

@@ -80,7 +80,6 @@ To facilitate the interplay between an `AbsProjectiveScheme` and the affine char
 `covered_scheme` we provide the following methods:
 ```@docs
     dehomogenization_map(X::AbsProjectiveScheme, U::AbsSpec)
-    dehomogenization_map(X::AbsProjectiveScheme, i::Int)
     homogenization_map(P::AbsProjectiveScheme, U::AbsSpec)
 ```
 

experimental/Experimental.jl

@@ -78,3 +78,4 @@ include("Schemes/ToricBlowups/attributes.jl")
 include("Schemes/ToricBlowups/methods.jl")
 
 include("ExteriorAlgebra/ExteriorAlgebra.jl")
+

experimental/PlaneCurve/docs/src/divisors.md

@@ -1,5 +1,5 @@
 ```@meta
-CurrentModule = Oscar
+CurrentModule = Oscar.PlaneCurveModule
 ```
 
 # Divisors

experimental/PlaneCurve/docs/src/elliptic_curves.md

@@ -1,5 +1,5 @@
 ```@meta
-CurrentModule = Oscar
+CurrentModule = Oscar.PlaneCurveModule
 DocTestSetup = quote
   using Oscar
 end
@@ -43,7 +43,7 @@ Addition and multiplication by an integer of a point on an elliptic curve can be
 
 #### Example
 
-```jldoctest
+```julia
 julia> T, (x, y, z) = graded_polynomial_ring(QQ, ["x", "y", "z"])
 (Graded multivariate polynomial ring in 3 variables over QQ, MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}[x, y, z])
 
@@ -107,7 +107,7 @@ rand_pair_EllCurve_Point(R::Oscar.MPolyDecRing{S}, PP::Oscar.Geometry.ProjSpc{S}
 
 Projective elliptic curves over a ring are for example used in the Elliptic Curve Method. We give here an example (see Example 7.1 of [Was08](@cite)) on how to use the previous functions to apply it. 
 
-```jldoctest
+```julia
 julia> n = 4453
 4453
 

experimental/PlaneCurve/docs/src/non_plane_curves.md

@@ -1,5 +1,5 @@
 ```@meta
-CurrentModule = Oscar
+CurrentModule = Oscar.PlaneCurveModule
 ```
 
 

experimental/PlaneCurve/docs/src/plane_curves.md

@@ -1,5 +1,5 @@
 ```@meta
-CurrentModule = Oscar
+CurrentModule = Oscar.PlaneCurveModule
 DocTestSetup = quote
   using Oscar
 end
@@ -72,7 +72,7 @@ in(P::Point{S}, C::AffinePlaneCurve{S}) where S <: FieldElem
 In order to define a point in the projective plane, one needs first to define
 the projective plane as follows, where `K` is the base ring:
 
-```jldoctest plane_curves
+```julia plane_curves
 julia> K = QQ
 Rational field
 
@@ -83,7 +83,7 @@ julia> PP = proj_space(K, 2)
 
 Then, one can define a projective point as follows:
 
-```jldoctest plane_curves
+```julia plane_curves
 julia> P = Oscar.Geometry.ProjSpcElem(PP[1], [QQ(1), QQ(2), QQ(-5)])
 (1 : 2 : -5)
 

experimental/PlaneCurve/src/AffinePlaneCurve.jl

@@ -29,7 +29,7 @@ export tangent_lines
 Throw an error if `P` is not a point of `C`, return `false` if `P` is a singular point of `C`, and `true` if `P` is a smooth point of `C`.
 
 # Examples
-```jldoctest
+```julia
 julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
 (Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
 
@@ -63,7 +63,7 @@ end
 Return the tangent of `C` at `P` when `P` is a smooth point of `C`, and throw an error otherwise.
 
 # Examples
-```jldoctest
+```julia
 julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
 (Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
 
@@ -102,7 +102,7 @@ end
 Return the affine plane curve consisting of the common component of `C` and `D`, or an empty vector if they do not have a common component.
 
 # Examples
-```jldoctest
+```julia
 julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
 (Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
 
@@ -138,7 +138,7 @@ end
 Return a list whose first element is the affine plane curve defined by the gcd of `C.eq` and `D.eq`, the second element is the list of the remaining intersection points when the common components are removed from `C` and `D`.
 
 # Examples
-```jldoctest
+```julia
 julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
 (Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
 
@@ -231,14 +231,14 @@ end
 Return the reduced singular locus of `C` as a list whose first element is the affine plane curve consisting of the singular components of `C` (if any), and the second element is the list of the isolated singular points (which may be contained in the singular component). The singular component might not contain any point over the considered field.
 
 # Examples
-```jldoctest
+```julia
 julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
 (Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
 
-julia> C = AffinePlaneCurve(x^2*(x+y)*(y^3-x^2))
+julia> C = PlaneCurveModule.AffinePlaneCurve(x^2*(x+y)*(y^3-x^2))
 Affine plane curve defined by -x^5 - x^4*y + x^3*y^3 + x^2*y^4
 
-julia> curve_singular_locus(C)
+julia> PlaneCurveModule.curve_singular_locus(C)
 2-element Vector{Vector}:
  AffinePlaneCurve[Affine plane curve defined by x]
  Point[Point with coordinates QQFieldElem[-1, 1], Point with coordinates QQFieldElem[0, 0]]
@@ -317,17 +317,17 @@ end
 Return the multiplicity of `C` at `P`.
 
 # Examples
-```jldoctest
+```julia
 julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
 (Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
 
-julia> C = AffinePlaneCurve(x^2*(x+y)*(y^3-x^2))
+julia> C = PlaneCurveModule.AffinePlaneCurve(x^2*(x+y)*(y^3-x^2))
 Affine plane curve defined by -x^5 - x^4*y + x^3*y^3 + x^2*y^4
 
-julia> P = Point([QQ(2), QQ(-2)])
+julia> P = PlaneCurveModule.Point([QQ(2), QQ(-2)])
 Point with coordinates QQFieldElem[2, -2]
 
-julia> multiplicity(C, P)
+julia> PlaneCurveModule.multiplicity(C, P)
 1
 ```
 """
@@ -353,17 +353,17 @@ end
 Return the tangent lines at `P` to `C` with their multiplicity.
 
 # Examples
-```jldoctest
+```julia
 julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
 (Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
 
-julia> C = AffinePlaneCurve(x^2*(x+y)*(y^3-x^2))
+julia> C = PlaneCurveModule.AffinePlaneCurve(x^2*(x+y)*(y^3-x^2))
 Affine plane curve defined by -x^5 - x^4*y + x^3*y^3 + x^2*y^4
 
-julia> P = Point([QQ(0), QQ(0)])
+julia> P = PlaneCurveModule.Point([QQ(0), QQ(0)])
 Point with coordinates QQFieldElem[0, 0]
 
-julia> tangent_lines(C, P)
+julia> PlaneCurveModule.tangent_lines(C, P)
 Dict{AffinePlaneCurve{QQFieldElem}, Int64} with 2 entries:
   x     => 4
   x + y => 1
@@ -418,20 +418,20 @@ end
 Return the intersection multiplicity of `C` and `D` at `P`.
 
 # Examples
-```jldoctest
+```julia
 julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
 (Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
 
-julia> C = AffinePlaneCurve((x^2+y^2)*(x^2 + y^2 + 2*y))
+julia> C = PlaneCurveModule.AffinePlaneCurve((x^2+y^2)*(x^2 + y^2 + 2*y))
 Affine plane curve defined by x^4 + 2*x^2*y^2 + 2*x^2*y + y^4 + 2*y^3
 
-julia> D = AffinePlaneCurve((x^2+y^2)*(y^3*x^6 - y^6*x^2))
+julia> D = PlaneCurveModule.AffinePlaneCurve((x^2+y^2)*(y^3*x^6 - y^6*x^2))
 Affine plane curve defined by x^8*y^3 + x^6*y^5 - x^4*y^6 - x^2*y^8
 
-julia> Q = Point([QQ(0), QQ(-2)])
+julia> Q = PlaneCurveModule.Point([QQ(0), QQ(-2)])
 Point with coordinates QQFieldElem[0, -2]
 
-julia> intersection_multiplicity(C, D, Q)
+julia> PlaneCurveModule.intersection_multiplicity(C, D, Q)
 2
 ```
 """
@@ -454,26 +454,26 @@ end
 Return `true` if `C` and `D` intersect transversally at `P` and `false` otherwise.
 
 # Examples
-```jldoctest
+```julia
 julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
 (Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
 
-julia> C = AffinePlaneCurve(x*(x+y))
+julia> C = PlaneCurveModule.AffinePlaneCurve(x*(x+y))
 Affine plane curve defined by x^2 + x*y
 
-julia> D = AffinePlaneCurve((x-y)*(x-2))
+julia> D = PlaneCurveModule.AffinePlaneCurve((x-y)*(x-2))
 Affine plane curve defined by x^2 - x*y - 2*x + 2*y
 
-julia> P = Point([QQ(0), QQ(0)])
+julia> P = PlaneCurveModule.Point([QQ(0), QQ(0)])
 Point with coordinates QQFieldElem[0, 0]
 
-julia> Q = Point([QQ(2), QQ(-2)])
+julia> Q = PlaneCurveModule.Point([QQ(2), QQ(-2)])
 Point with coordinates QQFieldElem[2, -2]
 
-julia> aretransverse(C, D, P)
+julia> PlaneCurveModule.aretransverse(C, D, P)
 false
 
-julia> aretransverse(C, D, Q)
+julia> PlaneCurveModule.aretransverse(C, D, Q)
 true
 ```
 """
@@ -490,14 +490,14 @@ end
 Return `true` if `C` has no singular point, and `false` otherwise.
 
 # Examples
-```jldoctest
+```julia
 julia> R, (x, y) = polynomial_ring(QQ, ["x", "y"])
 (Multivariate polynomial ring in 2 variables over QQ, QQMPolyRingElem[x, y])
 
-julia> C = AffinePlaneCurve(x*(x+y))
+julia> C = PlaneCurveModule.AffinePlaneCurve(x*(x+y))
 Affine plane curve defined by x^2 + x*y
 
-julia> is_smooth_curve(C)
+julia> PlaneCurveModule.is_smooth_curve(C)
 false
 ```
 """
@@ -534,14 +534,14 @@ end
 Return the geometric genus of the projective closure of `C`.
 
 # Examples
-```jldoctest
+```julia
 julia> R, (x, y) = polynomial_ring(GF(7), ["x", "y"])
 (Multivariate polynomial ring in 2 variables over GF(7), fpMPolyRingElem[x, y])
 
-julia> C = AffinePlaneCurve(y^9 - x^2*(x-1)^9)
+julia> C = PlaneCurveModule.AffinePlaneCurve(y^9 - x^2*(x-1)^9)
 Affine plane curve defined by 6*x^11 + 2*x^10 + 6*x^9 + x^4 + 5*x^3 + x^2 + y^9
 
-julia> geometric_genus(C)
+julia> PlaneCurveModule.geometric_genus(C)
 0
 ```
 """