Quatroids and Rational Plane Cubics

TL;DR

This paper classifies the configurations of eight points in the plane using quatroids, providing a detailed stratification and insights into rational cubic curves passing through these points.

Contribution

It introduces quatroids as a new combinatorial tool to analyze point configurations and fully enumerates them for eight points, enhancing understanding of rational cubics in algebraic geometry.

Findings

Computed all 779,777 quatroids on eight points

Described the stratification of point configurations

Provided invariants and rationality certificates for cubics

Abstract

It is a classical result that there are $12$ (irreducible) rational cubic curves through $8$ generic points in $\mathbb{P}_{\mathbb{C}}^2$, 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.