Quatroids and Rational Plane Cubics

Taylor Brysiewicz; Fulvio Gesmundo; Avi Steiner

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arXiv · 2309.07357

Quatroids and Rational Plane Cubics

Taylor Brysiewicz·Fulvio Gesmundo·Avi Steiner

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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.

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BibTeX RIS

Taylor Brysiewicz, Fulvio Gesmundo, Avi Steiner. 2023-09-14. Quatroids and Rational Plane Cubics. https://arxiv.org/abs/2309.07357

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