Plane curves containing a star configuration

TL;DR

This paper studies the family of plane algebraic curves of degree d that contain a specific star configuration formed by pairwise intersections of general lines, providing dimension calculations for these families.

Contribution

It computes the dimension of the space of degree d curves containing a star configuration of points in the projective plane.

Findings

Dimension formulas for families of curves containing star configurations.

Results applicable to understanding the geometry of special point configurations.

Insights into the algebraic conditions defining such curves.

Abstract

Given a collection of $l$ general lines $\ell_1,\ldots,\ell_{l}$ in $\pr^2$, the star configuration $\XX(l)$ is the set of points constructed from all pairwise intersections of these lines. For each non-negative integer $d$, we compute the dimension of the family of curves of degree $d$ that contain a star configuration.