On the Sylvester-Gallai and the orchard problem for pseudoline arrangements

TL;DR

This paper classifies specific pseudoline arrangements with 19 triple points and 9 double points, explores their realizability with straight lines, and addresses related combinatorial geometry problems.

Contribution

It provides a complete classification of certain pseudoline arrangements and introduces new examples distinct from known configurations.

Findings

Classified pseudoline arrangements with 19 triple and 9 double points.

Identified arrangements that cannot be realized with straight lines.

Provided negative answers to Gr"unbaum's problems.

Abstract

We study a non-trivial extreme case of the orchard problem for $12$ pseudolines and we provide a complete classification of pseudoline arrangements having $19$ triple points and $9$ double points. We have also classified those that can be realized with straight lines. They include new examples different from the known example of B\"or\"oczky. Since Melchior's inequality also holds for arrangements of pseudolines, we are able to deduce that some combinatorial point-line configurations cannot be realized using pseudolines. In particular, this gives a negative answer to one of Gr\"unbaum's problems. We formulate some open problems which involve our new examples of line arrangements.