From Pappus Theorem to parameter spaces of some extremal line point configurations and applications

TL;DR

This paper investigates the parameter spaces of extremal line point configurations related to the Dirac-Motzkin Conjecture, revealing their geometric structures and implications in algebraic geometry and commutative algebra.

Contribution

It characterizes the parameter spaces of B"or"oczky configurations, showing one is a rational variety and the other is an elliptic curve with finitely many rational points.

Findings

The parameter space of B12 configurations is a three-dimensional rational variety.

The moduli space of B15 configurations is an elliptic curve with finitely many rational points.

No rational B15 configurations exist.

Abstract

In the present work we study parameter spaces of two line point configurations introduced by B\"or\"oczky. These configurations are extremal from the point of view of Dirac-Motzkin Conjecture settled recently by Green and Tao. They have appeared also recently in commutative algebra in connection with the containment problem for symbolic and ordinary powers of homogeneous ideals and in algebraic geometry in considerations revolving around the Bounded Negativity Conjecture. Our main results are Theorem A and Theorem B. We show that the parameter space of what we call $B12$ configurations is a three dimensional rational variety. As a consequence we derive the existence of a three dimensional family of rational $B12$ configurations. On the other hand the moduli space of $B15$ configurations is shown to be an elliptic curve with only finitely many rational points, all corresponding to degenerate configurations. Thus, somewhat surprisingly, we conclude that there are no rational $B15$ configurations.