On supersolvable and nearly supersolvable line arrangements

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...

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Bibliographic Details
Published inJournal of algebraic combinatorics Vol. 50; no. 4; pp. 363 - 378
Main Authors Dimca, Alexandru, Sticlaru, Gabriel
Format Journal Article
LanguageEnglish
Published New York Springer US 01.12.2019
Springer Nature B.V
Springer Verlag
Subjects
Algebraic Geometry
Combinatorial analysis
Combinatorics
Complex numbers
Computer Science
Convex and Discrete Geometry
Group Theory and Generalizations
Lattices
Mathematics
Mathematics and Statistics
Order
Ordered Algebraic Structures
Terao’s conjecture
32S22
Nearly free line arrangement
Tjurina number
Slope Problem
Jacobian syzygy
Free line arrangement
Primary 14H50
13D02
Secondary 14B05
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Summary: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.
ISSN:0925-9899
1572-9192
DOI:10.1007/s10801-018-0859-6