This repository contains Magma code for determining the stable reduction type of a quartic. The latter code works over number fields, including rationals (see [Bom+23b]).
An installation of Magma.
You can enable the functionality of this package in Magma by attaching the g3cayley/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/g3cayley/magma/spec");
The function QuarticTypeFromOctad(f, p) is the main function provided by this package. As input to a plane quartic curve f, it returns its stable reduction type modulo the prime number p (assumed to be greater than 2). In general, there are 74 possible types of stable reductions for a singular quartic: 42 of general type and 32 of hyperelliptic type.
For example, the following script returns (0eee), which means that modulo 3 the stable reduction of the curve consists of 3 elliptic curves all three secant to a curve of genus 0.
_<x,y,z> := PolynomialRing(Rationals(), 3);
f := 54*x^4 + 18*x^3*y + 18*x^3*z + 2*x^2*y^2 + 2*x^2*y*z + 8*x^2*z^2 + 18*x*y^3 +
2*x*y^2*z + 8*x*y*z^2 + 18*x*z^3 + 54*y^4 + 18*y^3*z + 8*y^2*z^2 + 18*y*z^3 + 54*z^4;
QuarticTypeFromOctad(f, 3);
The reduction modulo 7 of the same curve is a quartic with 2 nodes, i.e a curve of geometric genus 1 (type (1nn)), and modulo 37 a quartic with a single node, i.e a curve of geometric genus 2 (type (2n)).
QuarticTypeFromOctad(f, 7);
QuarticTypeFromOctad(f, 37);
Of independent interest, the function QuarticByReductionType(type, p) (resp. G3HyperReductionType(type, p) with G3QuarticFromHyper(h, p^2)) returns a quartic of the type given in input, among the 42 possible ones (resp. among the 32 possible hyperelliptic ones).
Examples are given in the directory examples. A full list of intrinsics is here.
Verbose comments are enabled by
SetVerbose("G3Cayley", n);
where n is either 0, 1 or 2. A higher value gives more comments.
Please cite the following preprints if this code has been helpful in your research:
[Bom+23] Raymond van Bommel, Jordan Docking, Vladimir Dokchitser, Reynald Lercier and Elisa Lorenzo García, Reduction of Plane Quartics and Cayley Octads, arXiv:2309.17381.
[Bom+24] Raymond van Bommel, Jordan Docking, Reynald Lercier and Elisa Lorenzo García, Reduction of Plane Quartics and Dixmier-Ohno invariant, arXiv:2401.xxxxx.