Quatroids and Rational Plane Cubics

Overview

This page contains code for verifying the claims made in the article:
Taylor Brysiewicz, Fulvio Gesmundo, Avi Steiner : Quatroids and rational plane cubics

Abstract. It is a classical result that there are 12 (irreducible) rational cubic curves through 8 generic points in the complex projective plane, but little is known about the non-generic cases. The space of 8-point configurations is partitioned into strata depending on combinatorial objects we call quatroids, a higher-order version of representable matroids. We compute all 779777 quatroids on eight distinct points in the plane, which produces a full description of the stratification. For each stratum, we generate several invariants, including the number of rational cubics through a generic configuration. As a byproduct of our investigation, we obtain a collection of results regarding the base loci of pencils of cubics and positive certificates for non-rationality.

Below, we showcase one illustration of a quatroid together with several of its invariants, in particular the number \(d_{\mathcal Q}\) of rational cubics through a generic configuration which exhibits the specificied linear and quadratic dependencies.

../_images/Q10.png

This page archives the code required to compute an exhaustive list of quatroids together with their invariants.

  • Quatroids.zip – Julia code auxiliary to the article. See the included README file for details.

  • QuatroidsM2Code.zip – Macaulay2 code auxiliary to the article. See the included README file for details.

  • Results.zip – The files generated by the julia and Macaulay2 code.

The following Jupyter notebook shows an example session where the results are produced. The notebook can also be downloaded here: QuatroidsJulia.ipynb.

The following two Jupyter notebooks show how to use the Quatroids julia package with examples. The notebooks can also be downloaded here: Quatroid66.ipynb, Quatroid97.ipynb.

Descriptions of the Auxiliary Files

../_images/QuatroidFiles.png

The Poset of Orbits of Bezoutian Quatroids

The following image can also be downloaded here: Poset.png

../_images/Poset.png

Illustrations of Candidate Quatroids

The following image can also be downloaded here: QuatroidsImage.png

../_images/QuatroidsImage.png

Project page created: 4/9/2023

Project contributors: Taylor Brysiewicz, Fulvio Gesmundo, Avi Steiner

Corresponding author of this page: Avi Steiner, avi.steiner@gmail.com

Macaulay2 code written by: Fulvio Gesmundo

Julia code written by: Taylor Brysiewicz

Jupyter notebooks written by: Avi Steiner

Software used: Julia (Version 1.10.4), Macaulay2 (v1.22)

System setup used: MacBook Pro with macOS 13.5.1 with Apple M2 Pro, 16 GB RAM

License for code of this project page: MIT License (https://spdx.org/licenses/MIT.html)

License for all other content of this project page (text, images, …): CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/)

Last updated 2/10/2024.