Introduction

This online help node and the nodes below it describe the specialized categories of nonsingular plane curves of genus zero: the rational plane curves and conics. Rational curves and conics in Magma are nonsingular plane curves of degree 1 and 2, respectively. The central functionality for conics concerns the existence of points over the rationals. If a point is known to exist, then a conic can be parametrized by a projective line or a rational curve. In addition several algorithms are implemented to convert conics to standard Legendre, or diagonal, models, and for curves over Q, to a reduced Legendre model or a minimal model. A rational curve in Magma is a linearly embedded image of the projective line, to which the full machinery of algebraic plane curves can be applied. The special category of conics is the called CrvCon and that of rational curves is CrvRat.

The central algorithms of this chapter deal with this classification and reduction of genus zero curves to one of the standard models to which efficient algorithms can be applied. These special types serve to classify all curves up to birational isomorphism. Since the canonical divisor K_C of a genus zero curve is of degree -2, a basis for the Riemann-Roch space of the effective divisor -K_C has dimension 3, and gives an anti-canonical embedding in the projective plane. The homogeneous quadratic relations between the functions define a conic model for any genus zero curve. If the curve has a rational point, then a similar construction with the divisor of this point gives a birational isomorphism with the projective line. For conics over the rationals, efficient algorithms of Cremona [CR02] allow one to first find a point, if one exists, and then to reduce to simpler models. The existence of a point is easily determined by local conditions, and this local data is carried by the data of the ramified or bad primes of reduction, and if such a point exists, the existence can be certified by a certificate. After a rational point is found, the curve can be parametrized by the projective line -- giving a birational isomorphism with the curve. In contrast to the algorithms for point finding, this reduction is independent of the base field.

Not every curve of genus zero can be "trivialized" by reduction to a rational curve in this way; the obstruction to having a rational point and therefore to being parametrized by a projective line is measured on the one hand by the primes of bad reduction, but also by the automorphism group, both of which are closely associated to an isomorphism class of quaternion algebras. The final algorithms of this chapter make use of this connection to compute the automorphism group of a curve, and to classify find isomorphisms between conics.

This chapter was written by David Kohel and Paulette Lieby of the University of Sydney, based on the former author's implementations of quaternion algebra arithmetic and an efficient implementation of Cremona's conics algorithms by the latter author. The reduction of genus zero curves to these standard models is based on the underlying algorithms of Florian Hess for Riemann-Roch spaces for function fields of curves.