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Article Dans Une Revue Rendiconti Lincei. Matematica e Applicazioni Année : 2019

Line and rational curve arrangements, and Walther's inequality

Alexandru Dimca
Gabriel Sticlaru
  • Fonction : Auteur

Résumé

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.

Dates et versions

hal-01761528 , version 1 (09-04-2018)

Identifiants

Citer

Alexandru Dimca, Gabriel Sticlaru. Line and rational curve arrangements, and Walther's inequality. Rendiconti Lincei. Matematica e Applicazioni, 2019, 30, pp.615--633. ⟨10.4171/RLM/863⟩. ⟨hal-01761528⟩
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