# Quatroids and Rational Plane Cubics

## Overview

**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.

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 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

## The Poset of Orbits of Bezoutian Quatroids

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

## Illustrations of Candidate Quatroids

The following image can also be downloaded here:
`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 1.9.3), 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 14/9/2023.