Parametrizing rational plane curves (#846)

Co-authored-by: Daniel Schultz <tthsqe12@gmail.com>

docs/doc.main

@@ -122,13 +122,15 @@
   ],
 
   "Algebraic Geometry" => [
-#     "Curves" => [
+     "AlgebraicGeometry/intro.md",
+     "Curves" => [
 #        "Experimental/introduction.md",
 #        "Experimental/plane_curves.md",
 #        "Experimental/divisors.md",
 #        "Experimental/elliptic_curves.md",
 #        "Experimental/non_plane_curves.md",
-#     ],
+	 "AlgebraicGeometry/para_rational_curves.md",	 
+     ],
      "Toric Varieties" => [
         "ToricVarieties/intro.md",
         "ToricVarieties/NormalToricVarieties.md",

docs/oscar_references.bib

@@ -23,6 +23,18 @@ @Article{BDEPS04
   url           = {https://doi.org/10.1093/qjmath/55.3.237}
 }
 
+@InCollection{BDLP17,
+  author        = {Böhm, Janko and Decker, Wolfram and Laplagne, Santiago
+                  and Pfister, Gerhard},
+  title         = {Local to global algorithms for the Gorenstein adjoint
+                  ideal of a curve},
+  booktitle     = {Algorithmic and experimental methods in algebra, geometry,
+                  and number theory},
+  pages         = {51--96},
+  publisher     = {Springer, Cham},
+  year          = {2017}
+}
+
 @InProceedings{BDLP19,
   title         = {Computing integral bases via localization and Hensel
                   lifting},
@@ -66,6 +78,14 @@ @Article{BKR20
   url           = {https://doi.org/10.1090/mcom/3546}
 }
 
+@MastersThesis{Bhm99,
+  author        = {Böhm, Janko},
+  title         = {Parametrisierung rationaler Kurven},
+  school        = {Universität Bayreuth},
+  type          = {Diploma Thesis},
+  year          = {1999}
+}
+
 @Book{CLS11,
   author        = {David A. {Cox} and John B. {Little} and Henry K.
                   {Schenck}},

docs/src/AlgebraicGeometry/intro.md

@@ -0,0 +1,21 @@
+```@meta
+CurrentModule = Oscar
+```
+
+```@setup oscar
+using Oscar
+```
+
+```@contents
+Pages = ["intro.md"]
+```
+
+# Introduction
+
+In this chapter, we introduce structures and functionality for dealing with varieties, sheaves, and schemes.
+
+This part of Oscar is at the very beginning of its development.
+
+We refer to the individual sections for references which provide details on theory and algorithms.
+
+

docs/src/AlgebraicGeometry/para_rational_curves.md

@@ -0,0 +1,95 @@
+```@meta
+CurrentModule = Oscar
+```
+
+```@setup oscar
+using Oscar
+```
+
+```@contents
+Pages = ["para_rational_curves.md"]
+```
+
+# Rational Parametrizations of Rational Plane Curves
+
+!!! note
+    In this section, $C$ will denote a complex projective plane curve, defined by an absolutely irreducible,
+    homogeneous polynomial in three variables, with coeffients in $\mathbb Q$. Moreover, we will write $n = \deg C$.
+
+Recall that the curve $C$ is *rational* if it is birationally equivalent to the projective line $\mathbb P^1(\mathbb C)$.
+In other words, there exists a *rational parametrization* of $C$, that is, a birational map $\mathbb P^1(\mathbb C)\dashrightarrow C$.
+Note that such a parametrization is given by three homogeneous polynomials of the same degree in the homogeneous coordinates on
+$\mathbb P^1(\mathbb C)$.
+
+!!! note
+    The curve $C$ is rational iff its geometric genus is zero.
+
+Based on work of Max Noether on adjoint curves, Hilbert und Hurwitz showed that if
+$C$ is rational, then there is a birational map $C \dashrightarrow D$ defined over $\mathbb Q$ such
+that $D = \mathbb P^1(\mathbb C)$ if $n$ is odd, and $D\subset\mathbb P^2(\mathbb C)$ is a conic if $n$ is even.
+
+!!! note
+    If a conic $D$ contains a rational point, then there exists a parametrization of $D$ defined over $\mathbb Q$;
+    otherwise, there exists a parametrization of $D$ defined over a quadratic field extension of $\mathbb Q$.
+
+The approach of Hilbert und Hurwitz is constructive and allows one, in principle, to find rational parametrizations.
+The resulting algorithm is not very practical, however, as the approach asks to compute adjoint curves repeatedly,
+at each of a number of reduction steps.
+
+The algorithm implemented in OSCAR relies on reduction steps of a different type and requires the computation of adjoint
+curves only once. Its individual steps are interesting in their own right:
+
+ - Assure that the curve  $C$ is rational by checking that its geometric genus is zero;
+ - compute a basis of the adjoint curves of $C$ of degree ${n-2}$; each such basis defines a birational map $C \dashrightarrow C_{n-2},$
+    where $C_{n-2}$ is a rational normal curve in $\mathbb P^{n-2}(\mathbb C)$;
+ - the anticanonical linear system on $C_{n-2}$ defines a birational map $C_{n-2}\dashrightarrow C_{n-4}$, where $C_{n-4}$ is a rational normal curve in in $\mathbb P^{n-4}(\mathbb C)$;
+ - iterate the previous step to obtain a birational map  $C_{n-2} \dashrightarrow \dots \dashrightarrow D$,
+    where $D = \mathbb P^1(\mathbb C)$ if $n$ is odd, and $D\subset\mathbb P^2(\mathbb C)$ is a conic if $n$ is even;
+ - invert the birational map  $C \dashrightarrow C_{n-2} \dashrightarrow \dots \dashrightarrow D$; 
+ - if $n$ is even, compute a parametrization of the conic $D$ and compose it with the inverted map above.
+
+!!! note
+    The defining property of an adjoint curve is that it passes with “sufficiently high” multiplicity through the singularities of $C$.
+    There are several concepts of making this precise. For each such concept, there is a corresponding  *adjoint ideal* of $C$,
+	namely the homogeneous ideal formed by the defining polynomials of the adjoint curves. In OSCAR, we follow
+	the concept of Gorenstein which leads to the largest possible adjoint ideal.
+
+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::MPolyElem{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{fmpq})
+```
+
+## Rational Points on Conics
+
+```@docs
+rational_point_conic(D::ProjPlaneCurve{fmpq})
+```
+## Parametrizing Rational Plane Curves
+
+```@docs
+ parametrization_plane_curve(C::ProjPlaneCurve{fmpq})
+```
+
+
+
+
+
+

experimental/PlaneCurve/AffinePlaneCurve.jl

@@ -104,7 +104,7 @@ julia> D = Oscar.AffinePlaneCurve(x*(x+y)*(x-y))
 Affine plane curve defined by x^3 - x*y^2
 
 julia> Oscar.common_components(C, D)
-1-element Vector{Oscar.PlaneCurveModule.AffinePlaneCurve{fmpq}}:
+1-element Vector{AffinePlaneCurve{fmpq}}:
  Affine plane curve defined by x^2 + x*y
 ```
 """
@@ -141,8 +141,8 @@ Affine plane curve defined by x^2 - x*y - 2*x + 2*y
 
 julia> Oscar.curve_intersect(C, D)
 2-element Vector{Vector{T} where T}:
- Oscar.PlaneCurveModule.AffinePlaneCurve[]
- Oscar.PlaneCurveModule.Point{fmpq}[Point with coordinates fmpq[0, 0], Point with coordinates fmpq[2, -2]]
+ AffinePlaneCurve[]
+ Point{fmpq}[Point with coordinates fmpq[0, 0], Point with coordinates fmpq[2, -2]]
 ```
 """
 function curve_intersect(C::AffinePlaneCurve{S}, D::AffinePlaneCurve{S}) where S <: FieldElem
@@ -231,8 +231,8 @@ Affine plane curve defined by -x^5 - x^4*y + x^3*y^3 + x^2*y^4
 
 julia> Oscar.curve_singular_locus(C)
 2-element Vector{Vector{T} where T}:
- Oscar.PlaneCurveModule.AffinePlaneCurve[Affine plane curve defined by x]
- Oscar.PlaneCurveModule.Point[Point with coordinates fmpq[-1, 1], Point with coordinates fmpq[0, 0]]
+ AffinePlaneCurve[Affine plane curve defined by x]
+ Point[Point with coordinates fmpq[-1, 1], Point with coordinates fmpq[0, 0]]
 ```
 """
 function curve_singular_locus(C::AffinePlaneCurve)
@@ -355,7 +355,7 @@ julia> P = Oscar.Point([QQ(0), QQ(0)])
 Point with coordinates fmpq[0, 0]
 
 julia> Oscar.tangent_lines(C, P)
-Dict{Oscar.PlaneCurveModule.AffinePlaneCurve{fmpq}, Int64} with 2 entries:
+Dict{AffinePlaneCurve{fmpq}, Int64} with 2 entries:
   x     => 4
   x + y => 1
 ```