Seshadri constants of singular loci of line arrangements with at most twelve lines

Abstract

Let ${\mathcal L}$ be an arrangement of $d$ lines in the complex projective plane and let $Z={\operatorname{Sing}}({\mathcal L})$. A question of Pokora asks whether the multipoint Seshadri constant of ${\mathcal O}_{\mathbb{P}^2}(1)$ at $Z$ is always computed by a line of the arrangement containing the largest number of points of $Z$. We prove that the answer is affirmative for every arrangement of at most twelve lines. The proof uses a Bézout-type covering criterion, elementary restrictions on the intersection lattice, and a detailed analysis of the two extremal combinatorial types that can occur for twelve lines. One type is the Hesse arrangement; it is handled using a net of quartics through its twelve double points. The other type would yield a uniquely determined incidence structure on thirteen lines. We show that this incidence structure is not realizable over any field.