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Quatroids and Rational Plane Cubics

Author:
Taylor Brysiewicz, Fulvio Gesmundo, Avi Steiner
Keyword:
Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG)
journal:
--
date:
2023-09-13 16:00:00
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.
PDF: Quatroids and Rational Plane Cubics.pdf
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