Generalization of the addition and restriction theorems from free arrangements to the class of projective dimension one
Algebraic Combinatorics, Volume 7 (2024) no. 2, pp. 413-429.

We study a generalized version of Terao’s addition theorem for free arrangements to the category of those with projective dimension one. Namely, we give a formulation to determine the algebraic structure of the logarithmic derivation module of a hyperplane arrangement obtained by adding one hyperplane to a free arrangement under the assumption that the arrangement obtained by restricting onto that hyperplane is free too.

Also, we introduce a class of stair-strictly plus-one generated (SPOG) arrangements whose SPOGness depends only on the intersection lattice similar to the class of stair-free arrangements which satisfies Terao’s conjecture.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.340
Classification: 32S22, 52C35
Keywords: hyperplane arrangements, logarithmic derivation modules, free arrangements, SPOG arrangements, the addition-deletion theorems
Abe, Takuro 1

1 Rikkyo University Department of Mathematics 3-34-1 Nishi-Ikebukuro, Toshima-Ku 1718501, Tokyo Japan
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{ALCO_2024__7_2_413_0,
     author = {Abe, Takuro},
     title = {Generalization of the addition and restriction theorems from free arrangements to the class of projective dimension one},
     journal = {Algebraic Combinatorics},
     pages = {413--429},
     publisher = {The Combinatorics Consortium},
     volume = {7},
     number = {2},
     year = {2024},
     doi = {10.5802/alco.340},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.340/}
}
TY  - JOUR
AU  - Abe, Takuro
TI  - Generalization of the addition and restriction theorems from free arrangements to the class of projective dimension one
JO  - Algebraic Combinatorics
PY  - 2024
SP  - 413
EP  - 429
VL  - 7
IS  - 2
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.340/
DO  - 10.5802/alco.340
LA  - en
ID  - ALCO_2024__7_2_413_0
ER  - 
%0 Journal Article
%A Abe, Takuro
%T Generalization of the addition and restriction theorems from free arrangements to the class of projective dimension one
%J Algebraic Combinatorics
%D 2024
%P 413-429
%V 7
%N 2
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.340/
%R 10.5802/alco.340
%G en
%F ALCO_2024__7_2_413_0
Abe, Takuro. Generalization of the addition and restriction theorems from free arrangements to the class of projective dimension one. Algebraic Combinatorics, Volume 7 (2024) no. 2, pp. 413-429. doi : 10.5802/alco.340. https://alco.centre-mersenne.org/articles/10.5802/alco.340/

[1] Abe, Takuro Divisionally free arrangements of hyperplanes, Invent. Math., Volume 204 (2016) no. 1, pp. 317-346 | DOI | MR | Zbl

[2] Abe, Takuro Deletion theorem and combinatorics of hyperplane arrangements, Math. Ann., Volume 373 (2019) no. 1-2, pp. 581-595 | DOI | MR | Zbl

[3] Abe, Takuro Projective dimensions of hyperplane arrangements, 2020 | arXiv

[4] Abe, Takuro Plus-one generated and next to free arrangements of hyperplanes, Int. Math. Res. Not. IMRN (2021) no. 12, pp. 9233-9261 | DOI | MR | Zbl

[5] Abe, Takuro Addition-deletion theorem for free hyperplane arrangements and combinatorics, J. Algebra, Volume 610 (2022), pp. 1-17 | DOI | MR

[7] Abe, Takuro; Dimca, Alexandru Splitting types of bundles of logarithmic vector fields along plane curves, Internat. J. Math., Volume 29 (2018) no. 8, Paper no. 1850055, 20 pages | DOI | MR | Zbl

[8] Grayson, Daniel R.; Stillman, Michael E. Macaulay2, a software system for research in algebraic geometry, Available at http://www2.macaulay2.com

[9] Orlik, Peter; Terao, Hiroaki Arrangements of hyperplanes, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 300, Springer-Verlag, Berlin, 1992, xviii+325 pages | DOI | MR

[10] Sakai, Y Graphs and plus-one generated arrangements of hyperplanes, Masters thesis, Kyushu University (2020) (in Japanese, 34 pages)

[11] Terao, Hiroaki Arrangements of hyperplanes and their freeness. I, II., J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 27 (1980) no. 2, pp. 293-320 | MR | Zbl

[12] Terao, Hiroaki Generalized exponents of a free arrangement of hyperplanes and Shepherd-Todd-Brieskorn formula, Invent. Math., Volume 63 (1981) no. 1, pp. 159-179 | DOI | MR

[13] Yoshinaga, Masahiko Characterization of a free arrangement and conjecture of Edelman and Reiner, Invent. Math., Volume 157 (2004) no. 2, pp. 449-454 | DOI | MR

[14] Yoshinaga, Masahiko Freeness of hyperplane arrangements and related topics, Ann. Fac. Sci. Toulouse Math. (6), Volume 23 (2014) no. 2, pp. 483-512 | DOI | MR | Zbl

[15] Ziegler, Günter M. Combinatorial construction of logarithmic differential forms, Adv. Math., Volume 76 (1989) no. 1, pp. 116-154 | DOI | MR

[16] Ziegler, Günter M. Multiarrangements of hyperplanes and their freeness, Singularities (Iowa City, IA, 1986) (Contemp. Math.), Volume 90, Amer. Math. Soc., Providence, RI, 1989, pp. 345-359 | DOI | MR | Zbl

Cited by Sources: