The general tools for constructing and analysing curves are described in Chapter ALGEBRAIC CURVES. We do not repeat them here, but rather give some examples in Section Rational Curve and Conic Examples to demonstrate those basics that the user will need. In this section we describe the main parametrisation function for rational curves and functions which enable type change from a curve of genus zero to a rational curve.
Rational curves and conics are the specialised types for nonsingular plane curves of genus zero defined by polynomials of degree 1 and 2, respectively. The condition of nonsingularity is equivalent to that of absolute irreducibility for conics, and imposes no condition on a linear equation in the plane.
Ambient: Sch Default:
This creates a conic curve with the given sequence of coefficients (which should have length 3 or 6). A sequence [a, b, c] designates the conic aX2 + bY2 + cZ2, while a sequence [a, b, c, d, e, f] designates the conic aX2 + bY2 + cZ2 + dXY + eYZ + fXZ.The optional parameter Ambient may be used to give the specific ambient projective space in which to create the conic; otherwise a new ambient will be created for it to lie in.
Ambient: Sch Default:
This creates a conic curve associated to M, which must be a symmetric 3 x 3 matrix. Explicitly, the equation of the conic is [X, Y, Z]M[X, Y, Z]tr.The optional parameter Ambient may be used to give the specific ambient projective space in which to create the conic; otherwise a new ambient will be created for it to lie in.
Returns the conic defined by the polynomial f in the projective plane X.
Returns true if and only if the scheme S is a nonsingular plane curve of degree 2, in which case it also returns a curve (of type CrvCon) with the same defining polynomial as S.
Returns the rational curve defined by the linear polynomial f in the projective plane X.
Returns true if and only if the scheme S is defined by a linear polynomial in some projective plane P2; if so, it also returns a curve of type CrvRat with the same defining polynomial as S in P2.
> P2<x,y,z> := ProjectivePlane(Rationals()); > C0 := Curve(P2, x^2 + 3*x*y + 2*y^2 - z^2); > C0; Curve over Rational Field defined by x^2 + 3*x*y + 2*x^2 - z^2 > bool, C1 := IsConic(C0);This creates a new curve C1 in the same ambient space P2, explicitly as a conic curve.
> C1; Conic over Rational Field defined by x^2 + 3*x*y + 2*y^2 - z^2 > AmbientSpace(C0) eq AmbientSpace(C1); true > DefiningIdeal(C0) eq DefiningIdeal(C1); true > Type(C0), Type(C1); CrvPln CrvCon
The basic access functions for rational curves and conics are inherited from the general machinery for plane curves and hypersurface schemes.
Returns the base ring of the curve C.
Returns the category of rational curves CrvRat or of conics CrvCon; these are special subtypes of planes curves (type CrvPln), which are themselves subtypes of general curves (type Crv).
Returns the defining polynomial of the conic or rational curve C.
Returns the defining ideal of the conic or rational curve C.
These examples illustrate how to obtain standard models of a curve of genus zero, either as a conic or as a parametrisation by the projective line.
We begin with an example of a singular curve of geometric genus zero.
> P2<x,y,z> := ProjectivePlane(FiniteField(71)); > C := Curve(P2, (x^3 + y^2*z)^2 - x^5*z); > C; Curve over GF(71) defined by x^6 + 70*x^5*z + 2*x^3*y^2*z + y^4*z^2 > ArithmeticGenus(C); 10 > Genus(C); 0 > #RationalPoints(C); 73 > Z := SingularSubscheme(C); > Degree(Z); 18We see that C is highly singular and that its desingularisation has genus zero. At most 18 of 73 points are singular; note that a nonsingular curve of genus zero would have 72 points. We now investigate the source of the extra points.
> cmps := IrreducibleComponents(Z);
> [ Degree(X) : X in cmps ];
[ 11, 7 ]
> [ Degree(ReducedSubscheme(X)) : X in cmps ];
[ 1, 1 ]
> [ RationalPoints(X) : X in cmps ];
[
{@ (0 : 0 : 1) @},
{@ (0 : 1 : 0) @}
]
Since the only singular rational points on C are (0 : 0 : 1) and
(0 : 1 : 0), the "obvious" point (1 : 0 : 1) must be nonsingular
and we can use it to obtain a rational parametrisation of the curve
as explained in Section Isomorphisms.
> P1<u,v> := ProjectiveSpace(FiniteField(71), 1);
> p := C![1, 0, 1];
> m := Parametrization(C, Place(p), Curve(P1));
> S1 := {@ m(q) : q in RationalPoints(P1) @};
> #S1;
72
> [ q : q in RationalPoints(C) | q notin S1 ];
[ (0 : 1 : 0) ]
We conclude that the extra point comes from a singularity
whose resolution does not have any degree one places over it
(see Section Divisors of Chapter ALGEBRAIC CURVES for background
on places of curves). We can verify this explicitly.
> [ Degree(p) : p in Places(C![0, 1, 0]) ]; [ 2 ]
> P2<x,y,z> := ProjectiveSpace(Rationals(), 2); > C0 := Curve(P2, x^2 - 54321*x*y + y^2 - 97531*z^2); > IsNonsingular(C0); trueThe curve C0 is defined as a degree 2 curve over the rationals. By making a preliminary type change to the type of conics, CrvCon, we can test whether there exists a rational point over Q and use efficient algorithms of Section Finding Points for finding rational points on curves in conic form. The existence of a point (defined over the base field) is equivalent to the existence of a parametrisation (defined over the base field) of the curve by the projective line.
> bool, C1 := IsConic(C0); > bool; true > C1; Conic over Rational Field defined by x^2 - 54321*x*y + y^2 - 97531*z^2 > HasRationalPoint(C1); true (398469/162001 : -118246/162001 : 1) > RationalPoint(C1); (398469/162001 : -118246/162001 : 1)The parametrisation intrinsic requires a one-dimensional ambient space as one of the arguments. This space will be used as the domain of the parametrisation map.
> P1<u,v> := ProjectiveSpace(Rationals(), 1); > phi := Parametrization(C1, Curve(P1)); > phi; Mapping from: Prj: P1 to CrvCon: C1 with equations : 398469*u^2 + 944072*u*v + 559185*v^2 -118246*u^2 - 200850*u*v - 85289*v^2 162001*u^2 + 329499*u*v + 162991*v^2 and inverse -4634102139*x + 30375658963*y + 31793350442*z 5456130497*x - 25641668406*y - 32136366969*z and alternative inverse equations : 5456130497*x - 25641668406*y - 32136366969*z -6424037904*x + 21645471041*y + 31600239062*zThe defining functions for the parametrisation may look large, but they are defined simply by a linear change of variables from the 2-uple embedding of the projective line in the projective plane. We now do a naive search for rational points on the curve.
> time RationalPoints(C1 : Bound := 100000);
{@ @}
Time: 2.420
Although there were no points with small coefficients, the
parametrisation provides us with any number of rational points:
> phi(P1![0, 1]); (559185/162991 : -85289/162991 : 1) > phi(P1![1, 1]); (1901726/654491 : -404385/654491 : 1) > phi(P1![1, 0]); (398469/162001 : -118246/162001 : 1)
> P2<x,y,z> := ProjectiveSpace(RationalField(), 2); > f := 1134*x^2 - 28523*x*y - 541003*x*z - 953*y^2 - 3347*y*z - 245*z^2; > C := Conic(P2, f); > LegendrePolynomial(C); 1134*x^2 - 927480838158*y^2 - 186042605936505203884941*z^2 > ReducedLegendrePolynomial(C); 817884337*x^2 - y^2 - 353839285266278*z^2Now we demonstrate how to extend the base field of the conic; that is, to create the conic with the same coefficients but in a larger field. This can be done by calling BaseExtend; in this instance, we move from the rationals to a particular number field.
(Note the assignment of names to C in order to make the printing nicer.)
> P<t> := PolynomialRing(RationalField()); > K := NumberField(t^2 - t + 723); > C<u,v,w> := BaseExtend(C, K); > C; Conic over K defined by 1134*u^2 - 28523*u*v - 953*v^2 - 541003*u*w - 3347*v*w - 245*w^2