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Description

This repository contains Magma code for reconstructing plane quartic curves from their Dixmier--Ohno invariants.

Prerequisites

You need to have Magma installed on your machine to use this code. These algorithms use the Magma algorithm MinimizeReducePlaneQuartic, due to Elsenhans and Stoll, to simplify any output over the rationals. In order for this to work properly, one needs a bug fix of the file magma/package/Geometry/SrfDP/, which can be found in BugFix.m. Alternatively, it is possible to attach this new file separately.

Installation

You can enable the functionality of this package in Magma by attaching the quartic_reconstruction/magma/spec file with AttachSpec. To make this independent of the directory in which you find yourself, and to active this on startup by default, you may want to indicate the relative path in your ~/.magmarc file, by adding the line

AttachSpec("~/Programs/quartic_reconstruction/magma/spec");

Usage

An example that tests the routines in this package can be found in Example.m.

Verbose comments are enabled by

SetVerbose("Reconstruction", n);

where n is either 1 or 2. A higher value gives more comments.

Credits

The papers described the algorithms for minimization and reduction algorithms that are mentioned above and that play a crucial role when using these algorithms over the rationals are:

Andreas-Stephan Elsenhans
Good models for cubic surfaces
Preprint at https://math.uni-paderborn.de/fileadmin/mathematik/AG-Computeralgebra/Preprints-elsenhans/red_5.pdf

Michael Stoll
Reduction theory of point clusters in projective space
Groups Geom. Dyn. 5 (2011), no. 2, 553-565
Preprint at arXiv:0909.2808

Citing this code

Please cite the following preprint if this code has been helpful in your research:

Reynald Lercier, Christophe Ritzenthaler, and Jeroen Sijsling
Reconstructing plane quartic from their invariants
Preprint at arXiv:1606.05594

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A Magma package for reconstructing plane quartics from Dixmier-Ohno invariants

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