Line and rational curve arrangements, and Walther’s inequality

Abstract

There are two invariants associated to any line arrangement: the freeness defect $\nu (C)$ and an upper bound for it, denoted by ${\nu }^{\prime }(C)$, coming from a recent result by Uli Walther. We show that ${\nu }^{\prime }(C)$ 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 $\nu (C)={\nu }^{\prime }(C)$ 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.