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.

Revised:

Accepted:

Published online:

Keywords: hyperplane arrangements, logarithmic derivation modules, free arrangements, SPOG arrangements, the addition-deletion theorems

^{1}

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

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

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

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

[6] Deletion-Restriction for Logarithmic Forms on Multiarrangements, 2022 | arXiv

[7] 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] Macaulay2, a software system for research in algebraic geometry, Available at http://www2.macaulay2.com

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

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

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

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

[16] 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: *