Line and rational curve arrangements, and Walther's inequality

TL;DR

This paper investigates invariants of line arrangements, showing that Walther's upper bound is combinatorially determined for odd numbers of lines and proposing conjectures relating the invariants to arrangement properties.

Contribution

It proves Walther's invariant bound is combinatorially determined for odd line arrangements and introduces conjectures linking invariants to arrangement multiplicities.

Findings

Walther's upper bound is combinatorially determined for odd line arrangements

Conjecture that equality of invariants characterizes arrangements with specific multiplicity points

Extended Schenck's regularity result to singular rational curve arrangements

Abstract

There are two invariants associated to any line arrangement: the freeness defect $\nu(C)$ and an upper bound for it, denoted by $\nu'(C)$, coming from a recent result by Uli Walther. We show that $\nu'(C)$ is combinatorially determined, at least when the number of lines in $C$ is odd, while the same property is conjectural for $\nu(C)$. In addition, we conjecture that the equality $\nu(C)=\nu'(C)$ holds if and only if the essential arrangement $C$ of $d$ lines has either a point of multiplicity $d-1$, or has only double and triple points. We prove both conjectures in some cases, in particular when the number of lines is at most 10. We also extend a result by H. Schenck on the Castenuovo-Mumford regularity of line arrangements to arrangements of possibly singular rational curves.