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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| Published in | Journal of algebraic combinatorics Vol. 50; no. 4; pp. 363 - 378 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.12.2019
Springer Nature B.V Springer Verlag |
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| 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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| AbstractList | 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. Vall\`es. 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. |
| Author | Dimca, Alexandru Sticlaru, Gabriel |
| Author_xml | – sequence: 1 givenname: Alexandru orcidid: 0000-0001-9679-9870 surname: Dimca fullname: Dimca, Alexandru email: dimca@unice.fr organization: CNRS, LJAD, Université Côte d’Azur – sequence: 2 givenname: Gabriel surname: Sticlaru fullname: Sticlaru, Gabriel organization: Faculty of Mathematics and Informatics, Ovidius University |
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| Keywords | Terao’s conjecture 32S22 Nearly free line arrangement Tjurina number Slope Problem Jacobian syzygy Free line arrangement Primary 14H50 13D02 Secondary 14B05 |
| Language | English |
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| References | Dimca (CR6) 2017; 163 Ungar (CR24) 1982; 33 Anzis, Tohăneanu (CR2) 2016; 140 CR17 Dimca (CR7) 2017; 66 CR13 du Plessis, Wall (CR14) 1999; 126 Okonek, Schneider, Spindler (CR18) 1980 Saito (CR20) 1971; 14 Cook, Harbourne, Migliore, Nagel (CR3) 2018; 154 Abe, Dimca (CR1) 2018; 29 Orlik, Terao (CR19) 1992 Scott (CR21) 1970; 77 Yoshinaga (CR25) 2014; 23 Ziegler (CR26) 1989; 76 CR4 Dimca, Sticlaru (CR10) 2017; 30 CR8 Dimca, Sernesi (CR9) 2014; 1 Dimca, Sticlaru (CR11) 2017; 24 Dimca (CR5) 2017 Faenzi, Vallès (CR15) 2014; 90 Tohăneanu (CR22) 2014; 117 Dimca, Sticlaru (CR12) 2018; 54 Jambu, Terao (CR16) 1984; 52 Tohăneanu (CR23) 2016; 339 P Ungar (859_CR24) 1982; 33 B Anzis (859_CR2) 2016; 140 D Faenzi (859_CR15) 2014; 90 P Orlik (859_CR19) 1992 AA du Plessis (859_CR14) 1999; 126 859_CR17 A Dimca (859_CR5) 2017 859_CR8 A Dimca (859_CR11) 2017; 24 859_CR13 859_CR4 P Scott (859_CR21) 1970; 77 A Dimca (859_CR9) 2014; 1 A Dimca (859_CR10) 2017; 30 A Dimca (859_CR12) 2018; 54 C Okonek (859_CR18) 1980 A Dimca (859_CR6) 2017; 163 K Saito (859_CR20) 1971; 14 M Jambu (859_CR16) 1984; 52 T Abe (859_CR1) 2018; 29 Masahiko Yoshinaga (859_CR25) 2014; 23 D Cook (859_CR3) 2018; 154 SO Tohăneanu (859_CR23) 2016; 339 G Ziegler (859_CR26) 1989; 76 A Dimca (859_CR7) 2017; 66 SO Tohăneanu (859_CR22) 2014; 117 |
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| SubjectTerms | 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 |
| Title | On supersolvable and nearly supersolvable line arrangements |
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