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docs/src/AlgebraicGeometry/Curves/ParametrizationPlaneCurves.md
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| ```@meta | ||
| CurrentModule = Oscar | ||
| ``` | ||
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| # Rational Parametrizations of Rational Plane Curves | ||
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| !!! note | ||
| In this section, $C$ will denote a complex projective plane curve, defined by an absolutely irreducible, | ||
| homogeneous polynomial in three variables, with coefficients in $\mathbb Q$. Moreover, we will write $n = \deg C$. | ||
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| Recall that the curve $C$ is *rational* if it is birationally equivalent to the projective line $\mathbb P^1(\mathbb C)$. | ||
| In other words, there exists a *rational parametrization* of $C$, that is, a birational map $\mathbb P^1(\mathbb C)\dashrightarrow C$. | ||
| Note that such a parametrization is given by three homogeneous polynomials of the same degree in the homogeneous coordinates on | ||
| $\mathbb P^1(\mathbb C)$. | ||
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| !!! note | ||
| The curve $C$ is rational iff its geometric genus is zero. | ||
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| Based on work of Max Noether on adjoint curves, Hilbert und Hurwitz showed that if | ||
| $C$ is rational, then there is a birational map $C \dashrightarrow D$ defined over $\mathbb Q$ such | ||
| that $D = \mathbb P^1(\mathbb C)$ if $n$ is odd, and $D\subset\mathbb P^2(\mathbb C)$ is a conic if $n$ is even. | ||
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| !!! note | ||
| If a conic $D$ contains a rational point, then there exists a parametrization of $D$ defined over $\mathbb Q$; | ||
| otherwise, there exists a parametrization of $D$ defined over a quadratic field extension of $\mathbb Q$. | ||
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| The approach of Hilbert und Hurwitz is constructive and allows one, in principle, to find rational parametrizations. | ||
| The resulting algorithm is not very practical, however, as the approach asks to compute adjoint curves repeatedly, | ||
| at each of a number of reduction steps. | ||
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| The algorithm implemented in OSCAR relies on reduction steps of a different type and requires the computation of adjoint | ||
| curves only once. Its individual steps are interesting in their own right: | ||
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| - Assure that the curve $C$ is rational by checking that its geometric genus is zero; | ||
| - compute a basis of the adjoint curves of $C$ of degree ${n-2}$; each such basis defines a birational map $C \dashrightarrow C_{n-2},$ | ||
| where $C_{n-2}$ is a rational normal curve in $\mathbb P^{n-2}(\mathbb C)$; | ||
| - the anticanonical linear system on $C_{n-2}$ defines a birational map $C_{n-2}\dashrightarrow C_{n-4}$, where $C_{n-4}$ is a rational normal curve in in $\mathbb P^{n-4}(\mathbb C)$; | ||
| - iterate the previous step to obtain a birational map $C_{n-2} \dashrightarrow \dots \dashrightarrow D$, | ||
| where $D = \mathbb P^1(\mathbb C)$ if $n$ is odd, and $D\subset\mathbb P^2(\mathbb C)$ is a conic if $n$ is even; | ||
| - invert the birational map $C \dashrightarrow C_{n-2} \dashrightarrow \dots \dashrightarrow D$; | ||
| - if $n$ is even, compute a parametrization of the conic $D$ and compose it with the inverted map above. | ||
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| !!! note | ||
| The defining property of an adjoint curve is that it passes with “sufficiently high” multiplicity through the singularities of $C$. | ||
| There are several concepts of making this precise. For each such concept, there is a corresponding *adjoint ideal* of $C$, | ||
| namely the homogeneous ideal formed by the defining polynomials of the adjoint curves. In OSCAR, we follow | ||
| the concept of Gorenstein which leads to the largest possible adjoint ideal. | ||
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| See [Bhm99](@cite) and [BDLP17](@cite) for details and further references. | ||
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| ## Adjoint Ideals of Plane Curves | ||
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| ```@docs | ||
| adjoint_ideal(C::ProjectivePlaneCurve{QQField}) | ||
| ``` | ||
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| ## Rational Points on Conics | ||
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| ```@docs | ||
| rational_point_conic(D::ProjectivePlaneCurve{QQField}) | ||
| ``` | ||
| ## Parametrizing Rational Plane Curves | ||
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| ```@docs | ||
| parametrization(C::ProjectivePlaneCurve{QQField}) | ||
| ``` | ||
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| ## Contact | ||
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| Please direct questions about this part of OSCAR to the following people: | ||
| * [Janko Böhm](https://www.mathematik.uni-kl.de/~boehm/), | ||
| * [Wolfram Decker](https://math.rptu.de/en/wgs/agag/people/head/decker). | ||
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| You can ask questions in the [OSCAR Slack](https://www.oscar-system.org/community/#slack). | ||
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| Alternatively, you can [raise an issue on github](https://www.oscar-system.org/community/#how-to-report-issues). |
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