RandomCurvesOverVerySmallFiniteFields -- randomly chosen smooth canonical curves over small finite fields

Description

This package can be seen as a refined version of the RandomCanonicalCurves package, which catches all possible missteps in the constructions. The construction follows the unirationality proof of M_g for g<=14 and the article Matrix factorizations and families of curves of genus 15 for the genus g=15 case. Since a unirational parametrization of M_g is only a rational map, bad choices of parameters in the construction might end up in the indeterminacy locus or other undesired subloci. Since for example a hypersurface in characteristic 2 contains about 90% of the F_2-rational points (see A quick and dirty irreducibility Test for Multivariate Polynomials over F_q), a failure of the construction in the various steps is quite likely. We catch all possible missteps, and try again until success.

For g<=10 we construct the canonical curves via plane models.

For 10<g<14 the canonical curves are constructed via space models.

For g=14 the construction is based on Verra's proof of the unirationality of M_14 (see The unirationality of the moduli space of curves of genus ≤14 ).

The g=15 construction relies on matrix factorizations and is based on the Macaulay2 Package Matrix factorizations and families of curves of genus 15.

For g <=14, the methods used in this package are based on the Macaulay2 Package RandomCurves and the methods for the g=15 case are based on the Macaulay2-package MatFac15.

Caveat

This package requires Macaulay2 Version 1.9 or newer.

Authors

Version

This documentation describes version 0.3 of RandomCurvesOverVerySmallFiniteFields.

Citation

If you have used this package in your research, please cite it as follows:

@misc{RandomCurvesOverVerySmallFiniteFieldsSource,
  title = {{RandomCurvesOverVerySmallFiniteFields: A \emph{Macaulay2} package. Version~0.3}},
  author = {Christian Bopp and Frank-Olaf Schreyer},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Functions and commands
- isSmoothCurve -- Tests smoothness of a curve
- smoothCanonicalCurve -- Computes the ideal of canonical curve
- smoothCanonicalCurveGenus14 -- Compute a random canonical curve of genus 14
- smoothCanonicalCurveGenus15 -- Computes the ideal of canonical curve of genus 15
- smoothCanonicalCurveViaPlaneModel -- Computes the ideal of canonical curve via plane models
- smoothCanonicalCurveViaSpaceModel -- Computes the ideal of canonical curve via space models
Methods
- isSmoothCurve(Ideal) -- see isSmoothCurve -- Tests smoothness of a curve
- smoothCanonicalCurve(ZZ,ZZ) -- see smoothCanonicalCurve -- Computes the ideal of canonical curve
- smoothCanonicalCurveGenus14(ZZ) -- see smoothCanonicalCurveGenus14 -- Compute a random canonical curve of genus 14
- smoothCanonicalCurveGenus15(ZZ) -- see smoothCanonicalCurveGenus15 -- Computes the ideal of canonical curve of genus 15
- smoothCanonicalCurveViaPlaneModel(ZZ,ZZ) -- see smoothCanonicalCurveViaPlaneModel -- Computes the ideal of canonical curve via plane models
- smoothCanonicalCurveViaSpaceModel(ZZ,ZZ) -- see smoothCanonicalCurveViaSpaceModel -- Computes the ideal of canonical curve via space models
Symbols
- Details -- keeps track of the precise step in the construction of canonical genus 14 and 15 curves. Default value is "false".
- Printing -- displays information about the current step in the constructions. Default value is "false".

For the programmer

The object RandomCurvesOverVerySmallFiniteFields is a package, defined in RandomCurvesOverVerySmallFiniteFields.m2.

The source of this document is in RandomCurvesOverVerySmallFiniteFields.m2:1048:0.