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This online help node and the nodes below
it describe the specialised 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 parametrised 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 called
CrvCon and that of rational curves is
CrvRat.
The central algorithms of this chapter deal with the 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 KC of a genus zero curve
is of degree -2, a basis for the Riemann--Roch space of the effective
divisor -KC 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
Simon [Sim05] 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; if such a point exists then the existence
can be certified by a certificate.
Simon's algorithm in fact parametrises the curve (by the projective
line) which gives a birational isomorphism with the curve.
Not every curve of genus zero can be "trivialised" by reduction
to a rational curve in this way; the obstruction to having a rational
point, and therefore to being parametrised by a projective line, is
measured by the primes of bad reduction and 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 find isomorphisms
between conics.
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