In this section we discuss isomorphisms between heterogeneous types --- isomorphisms between combinations of curves of type Crv, CrvCon, and CrvRat, and their parametrisations by a projective line.
The first function which we treat here is Conic, which is as much a constructor as an isomorphism. It takes an arbitrary genus zero curve C and uses the anti-canonical divisor -KC of degree 2 to construct the Riemann--Roch space. For a genus zero curve this is a dimension 3 space of degree 2 functions and gives a projective embedding of C in P2 as a conic. This provides the starting point to make any genus zero curve amenable to the powerful machinery for point finding and isomorphism classification of conics.
On the other hand, if C is an arbitrary genus 0 curve with a known non-singular point or place p, then a parametrization of C can be constructed directly from the degree 1 Riemann--Roch space. There are intrinsics to do this.
An isomorphism --- provided that one exists --- of the projective line with a conic can be described as follows. The 2-uple embedding φ: P1 -> P2 defined by (u:v) |-> (u2:uv:v2) gives an isomorphism of P1 with the conic C0 with defining equation y2 = xz. The inverse isomorphism C0 -> P1 is defined by the maps
matrix(
(x:y:z) |-> (x:y) on x ≠0,
(x:y:z) |-> (y:z) on z ≠0, )
respectively. Since these open sets cover the conic C0 this defines an isomorphism and not just a birational map. In order to describe an isomorphism of a conic C1 with P1 it is then necessary and sufficient to give a change of variables which maps C0 = φ(P1) onto the conic C1. This matrix is called the parametrisation matrix and is stored with C1 once a rational point is found.
Given a parametrization over the rationals of an arbitrary genus 0 curve in ordinary projective space, a simpler parametrization can often be found using the intrinsic ImproveParametrization from the Scheme chapter of the handbook.
Given a curve of genus zero, returns a conic determined by the anti-canonical embedding of C.
> P2<x,y,z> := ProjectivePlane(FiniteField(71)); > C0 := Curve(P2, (x^3 + y^2*z)^2 - x^5*z); > C1, m := Conic(C0); > C1; Conic over GF(71) defined by x^2 + 70*x*z + y^2 > m : Minimal; (x : y : z) -> (x^3*y*z : 70*x^5 + x^4*z + 70*x^2*y^2*z : x^3*y*z + y^3*z^2)
This function is an optimised routine for parametrising a conic C defined over Z or Q. It returns a 3 x 3 matrix M which defines a parametrisation of C as a projective change of variables from the 2-uple embedding of a projective line in the projective plane; i.e., for a point (x0:y0:z0) on C, the point(x1:y1:z1) = (x0:y0:z0) M
satisfies the equation y12 = x1z1. Note that as usual in Magma the action of M is on the right and, consistently, the action of scheme maps is also on the right.
> P2<x,y,z> := ProjectiveSpace(Rationals(), 2); > C0 := Curve(P2, (x^3 + y^2*z)^2 - x^5*z); > C1, m := Conic(C0); > C1; Conic over Rational Field defined by x^2 - x*z + y^2The curve C1 has obvious points, such as (1 : 0 : 1), which Magma internally verifies without requiring an explicit user call to HasRationalPoint.
> ParametrizationMatrix(C1); [1 0 1] [0 1 0] [0 0 1] > Evaluate(DefiningPolynomial(C1), [x, y, x+z]); -x*z + y^2We note (as is standard in Magma) that the action of matrices, as with maps of schemes, is a right action on coordinates (x : y : z).
Given a conic curve C over a general field, these functions return a parametrisation as an isomorphism of schemes P -> C. Here P is a copy of a projective line; it may be specified as one of the arguments or a new projective line will be created. Note that it is now required that P is given (or created) as a curve rather than as an ambient space (as used to be permitted). This allows the immediate use of pullback/push-forward functionality for the parametrisation map.When a rational point or place is not specified as one of the arguments then the base field of C must be one of the kinds allowed in HasRationalPoint. If the conic has no rational points then an error results.
These functions are as above (see Parametrization), but use different algorithms.When C is a plane curve with only ordinary singularities (see subsection Ordinary Plane Curves) then a slightly different procedure is followed that relies less on the general function field machinery and tends to be faster and can produce nicer parametrisations. The variants ParametrizeOrdinaryCurve allow direct calls to these more specialised procedures. The I argument is the adjoint ideal of C (loc. cit.), which may be passed in if already computed.
The final function listed is slightly different; it applies only to rational normal curves. i.e., non-singular rational curves of degree d in ordinary d-dimensional projective space for d ≥1. For the sake of speed the irreducibility of C is not checked. The function uses adjoint maps to find either a line or conic parametrisation of C: If d is odd then an isomorphism from the projective line to C is returned, and if d is even then an isomorphism from a plane conic is returned. The method uses no function field machinery and can be much faster than the general function.
> k := FiniteField(101); > P2<x,y,z> := ProjectiveSpace(k, 2); > f := x^7 + 3*x^3*y^2*z^2 + 5*y^4*z^3; > C := Curve(P2, f); > Genus(C); 0In order to parametrise the curve C we need to find a nonsingular point on it, or at least a point of C over which there exists a unique degree one place; geometrically, such a point is one at which C has a cusp. To find such a point we invoke the intrinsic RationalPoints on C; since C is defined over a finite field this call returns an indexed set of all the points of C that are rational over its base field.
Having done the previous in the background, we demonstrate that the particular point (2 : 33 : 1) is such a nonsingular rational point.
> p := C![2,33,1]; > p; (2 : 33 : 1) > IsNonsingular(C, p); trueThe parametrisation function takes a projective line as the third argument; this will be used as the domain of the parametrisation map.
> P1<u,v> := ProjectiveSpace(k, 1);
> phi := Parametrization(C, Place(p), Curve(P1));
> phi;
Mapping from: Prj: P1 to Prj: P2
with equations :
2*u^7 + 5*u^6*v + 81*u^5*v^2 + 80*u^4*v^3 + 13*u^3*v^4
33*u^7 + 88*u^6*v + 90*u^5*v^2 + 73*u^4*v^3 + 25*u^3*v^4 +
83*u^2*v^5 + 72*u*v^6 + 24*v^7
u^7
Finally we confirm that the map really does parametrise the curve C.
Note that the map is normalised so that the point at infinity on the
projective line P1 maps to the prescribed point p.
> Image(phi); Scheme over GF(101) defined by x^7 + 3*x^3*y^2*z^2 + 5*y^4*z^3 > DefiningIdeal(Image(phi)) eq DefiningIdeal(C); true > phi(P1![1, 0]); (2 : 33 : 1)