On supersolvable and nearly supersolvable line arrangements

Abstract

We introduce a new class of line arrangements in the projective plane, called nearly supersolvable, and show that any arrangement in this class is either free or nearly free. More precisely, we show that the minimal degree of a Jacobian syzygy for the defining equation of the line arrangement, which is a subtle algebraic invariant, is determined in this case by the combinatorics. When such a line arrangement is nearly free, we discuss the splitting types and the jumping lines of the associated rank two vector bundle, as well as the corresponding jumping points, introduced recently by S. Marchesi and J. Vallès. As a by-product of our results, we get a version of the Slope Problem, valid over the real and the complex numbers as well.

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Correspondence to Alexandru Dimca.

Additional information

A. Dimca: This work has been supported by the French government, through the \(\mathrm{UCA}^{\mathrm{JEDI}}\) Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-01.

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Dimca, A., Sticlaru, G. On supersolvable and nearly supersolvable line arrangements. J Algebr Comb 50, 363–378 (2019). https://doi.org/10.1007/s10801-018-0859-6

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Keywords

  • Jacobian syzygy
  • Tjurina number
  • Free line arrangement
  • Nearly free line arrangement
  • Slope Problem
  • Terao’s conjecture

Mathematics Subject Classification

  • Primary 14H50
  • Secondary 14B05
  • 13D02
  • 32S22