On the Sylvester-Gallai and the orchard problem for pseudoline arrangements
Combinatorics
2023-01-10 v4 Algebraic Geometry
Abstract
We study a non-trivial extreme case of the orchard problem for pseudolines and we provide a complete classification of pseudoline arrangements having triple points and 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.
Keywords
Cite
@article{arxiv.1607.05864,
title = {On the Sylvester-Gallai and the orchard problem for pseudoline arrangements},
author = {Jürgen Bokowski and Piotr Pokora},
journal= {arXiv preprint arXiv:1607.05864},
year = {2023}
}
Comments
5 figures, 11 pages, to appear in Periodica Mathematica Hungarica