Line and rational curve arrangements, and Walther’s inequality

  • Alexandru Dimca

    Université Côte d’Azur, Nice, France
  • Gabriel Sticlaru

    Ovidius University, Constanţa, Romania
Line and rational curve arrangements, and Walther’s inequality cover

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Abstract

There are two invariants associated to any line arrangement: the freeness defect and an upper bound for it, denoted by , coming from a recent result by Uli Walther. We show that is combinatorially determined, at least when the number of lines in is odd, while the same property is conjectural for . In addition, we conjecture that the equality holds if and only if the essential arrangement of lines has either a point of multiplicity , 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.

Cite this article

Alexandru Dimca, Gabriel Sticlaru, Line and rational curve arrangements, and Walther’s inequality. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 30 (2019), no. 3, pp. 615–633

DOI 10.4171/RLM/863