Line and rational curve arrangements, and Walther's inequality

2018/03/14 by Dimca, Alexandru, Sticlaru, Gabriel
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1803.05386

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.

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