Omittable Lines

Abstract

Given a collection of n lines in the real projective plane, a line $\ell$ is said to be omittable if $\ell$ is free of ordinary points of intersection---in other words, if all the intersection points of $\ell$ with other lines from the collection come at the intersection of three or more lines. Given a collection of lines $\mathcal{L}$, denoting by $\mathcal{O(L)}$ the set of all omittable lines in the collection and by $g(\mathcal{L})$ the cardinality of $\mathcal{O(L)}$, we describe three infinite families of lines that can serve as $\mathcal{O(L)}$ for suitable $\mathcal{L}$ and also display a finite set of sporadic additional examples in which $\mathcal{O(L)}$ does not fall into any of the three families. We derive bounds on the size of $g(\mathcal{L})$ when $\mathcal{O(L)}$ falls into one of the three infinite families and weaker bounds for the more general case.