Line and rational curve arrangements, and Walther’s inequality

Abstract

There are two invariants associated to any line arrangement: the freeness defect ν(C) and an upper bound for it, denoted by ν ′ (C), coming from a recent result by Uli Walther. We show that ν ′ (C) is combinatorially determined, at least when the number of lines in C is odd, while the same property is conjectural for ν(C). In addition, we conjecture that the equality ν(C) = ν ′ (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.

On the minimal value of global Tjurina numbers for line arrangements

“…Hence in this case ν(C) ∈ {0, 1}, and the upper bound given by Corollary 1.8 is 1, hence this bound is sharp in this case. Another upper bound for the freeness defect ν(C) of a line arrangement is discussed in [15].…”