Parametrization -- Rational parametrization of rational curves and related computations

Overview:

Parametrization is a package to compute rational parametrizations of rational curves defined over ℚ.

Suppose C is a rational plane curve C of degree n defined over ℚ.

We use the package AdjointIdeal to compute the adjoint ideal of C. (The package exports also all functions available in AdjointIdeal, e.g., geometricGenus.)

The corresponding linear system maps the curve birationally to a rational normal curve in ℙ^n-2.

Iterating the anticanonical map we give a projection of the rational normal curve to ℙ¹ for n odd or to a conic C₂ in ℙ² for n even.

In the case that n is even we test for the existence of a rational point on the conic and if so give a rational parametrization of the conic.

By inverting the birational map of C to ℙ¹ or the conic we obtain a rational parametrization of C. If n is odd or C₂ has a rational point C is parametrized by ℙ¹ otherwise by C₂.

The package is work in progress, so there will be future improvements and more testing is necessary.

For more theoretical details see J. Boehm: Rational parametrization of rational curves, http://www.mathematik.uni-kl.de/~boehm/diplom%20janko%20boehm.pdf.

Key user functions:

parametrize -- This is the universal rational parametrization function, it works for plane rational curves, in particular conics, and rational normal curves.

testParametrization -- Test a parametrization.

rationalPointOnConic -- Test for a rational point on a conic and find it if it exists.

mapToRNC -- Map a plane rational curve to a rational normal curve.

Setup:

This package uses the package AdjointIdeal, so install this first.

Place the file Parametrization.m2 somewhere into the M2 search path and install the package by doing