Addition-deletion results for the minimal degree of logarithmic derivations of arrangements

Alexandru Dimca; Gabriel Sticlaru; Takuro Abe

REVIEW 2 major objections 2 minor 30 references

Addition-deletion results for the minimal degree of logarithmic derivations of arrangements

T0 review · 2 major / 2 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves a single rule for when deleting or adding a hyperplane changes the minimal logarithmic derivation degree by one.

desk verdict Useful addition-deletion result for r(A), but the extremal Tjurina-maximality theorems as printed define non-linear curves, not line arrangements. read the letter →

arxiv 1908.06885 v2 pith:CJFDZKT2 submitted 2019-08-19 math.AG math.CO

Takuro Abe , Alexandru Dimca , Gabriel Sticlaru This is my paper · ORCID

classification math.AGmath.CO MSC 14H5014B0513D0232S2252S35

keywords logarithmicderivationsminimaldegreehyperplanearrangementslineJacobianrelationsTjurinanumberaddition-deletiontheoremcombinatorialdetermination

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes an addition-deletion theorem for $r(A)$, the smallest degree of a nonzero logarithmic derivation of a hyperplane arrangement (equivalently, the smallest degree of a Jacobian relation). For $H\in A$ and $A'=A\setminus\{H\}$, it proves that $r(A)=r(A')+1$ whenever $r(A')

What carries the argument

The carrying object is the comparison of logarithmic derivation modules via the polynomial $B$-theorem: for $H\in A$, any derivation of $A'$ of degree less than $|A'|-|A^H|$ automatically extends to a derivation of $A$. The proof of the addition-deletion theorem also uses the Euler restriction map $\rho:D(A)\to D(A^H)$; because $r(A^H)>r(A')$, the image of a minimal derivation under $\rho$ must be a scalar multiple of the Euler derivation, and subtracting that multiple produces a derivation that vanishes on $H$ and factors through $\alpha_H$, giving a lower-degree derivation of $A'$. In the rank-three line case this mechanism reduces to the simple intersection count $|A_H|\ge r'+2$.

What would settle it

Look at the defining equations for $A_d$ and $B_d$: for $j\ge 1$ each factor $x^{2j}+y^{3j}-z$ has degree at least two, so its zero set is not a line; the claimed line arrangement does not exist as written. A corrected construction would need genuine linear factors, and a direct computation of $r(A_d)$ or $r(B_d)$ from the Jacobian syzygies of any corrected family would settle whether the asserted values $r=d-4$ and $r=d-3$ can be achieved.

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Extended reading notes

Core claim

The central claim is that the minimal degree of a logarithmic derivation behaves predictably under a single hyperplane deletion. Concretely, let $A$ be an essential hyperplane arrangement, $H\in A$, $A'=A\setminus\{H\}$, and let $r,r',r''$ denote the minimal degrees for $A$, $A'$, and the restriction $A^H$. Theorem 2.14 asserts that $r=r'+1$ whenever $r'<r''$. The proof takes a minimal derivation of $A'$ that also lies in $D_H(A)$, uses the Euler restriction to $A^H$ to eliminate the part proportional to the Euler derivation, and then factors the remainder by $\alpha_H$ to obtain a derivation of $A'$ of degree $r'-1$, a contradiction. In rank three the criterion reads: if the new line $H$ meets $A'$ in at least $r'+2$ points, then $r(A'\cup\{H\})=r'+1$. From this the paper derives that for line arrangements with only double and triple points, $r(A)$ is determined by the intersection lattice if and only if $n_3\le 5$, with counterexamples for every $n_3\ge 6$, and it constructs Tjurina maximal line arrangements in several ranges where existence was open.

Load-bearing premise

The extremal Tjurina-maximality results for $r=d-4$ and $r=d-3$ assume that the printed equations define line arrangements, but as printed the factors such as $x^{2j}+y^{3j}-z$ are not linear forms, so those constructions do not define line arrangements and those two theorems are unsupported as stated.

Editorial extensions

If this is right

For line arrangements with only double and triple points, $r(A)$ is determined by the combinatorics if and only if $n_3\le 5$; for every $n_3\ge 6$ there are pairs with the same intersection lattice and different $r(A)$.
Adding a generic line to any line arrangement raises $r$ by exactly one; adding a generic line through a point of maximal multiplicity raises $r$ by one unless $r'=d'-m'$, in which case $r$ stays equal to $r'$.
Tjurina maximal line arrangements exist for all pairs $(d,r)$ with $d\ge 4$ and $d/2\le r\le 2(d-1)/3$, and in the odd case $d-r$ odd up to $3(d-1)/4$; in particular every odd degree $d=2r-1\ge 7$ admits such an arrangement.
For free arrangements, deleting a hyperplane $H$ with $|A|-|A^H|$ equal to the second exponent $d_2$ lowers the second exponent by one, while otherwise it stays $d_2$.
Pairs of arrangements with identical combinatorics but different $r(A)$ can be constructed for any $d\ge 9$ and any number $n_3\ge 6$ of triple points.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The criterion suggests an inductive algorithm for $r(A)$: repeatedly delete a line whose intersection count with the rest exceeds $r+2$; in arrangements with very few triple points such a line is guaranteed, turning the computation into a purely combinatorial recursion. This is my inference, not a claim in the paper.
If the extremal families $A_d$ and $B_d$ can be repaired by replacing the non-linear factors with genuine linear forms, the same addition-deletion induction would prove the existence of maximal Tjurina line arrangements of types $(d,d-4)$ and $(d,d-3)$; as printed those existence theorems are not established.
The sharp cutoff $n_3=5$ likely reflects the incidence structure of triple points: with at most five triple points some line must contain a triple point that is alone on that line, which makes the deletion step applicable, whereas six triple points allow configurations in which every triple point shares its lines with others.
Because $r(A)$ and the generic splitting type of the logarithmic bundle determine each other when $r(A)<d/2$, the $n_3\le 5$ result transfers to a combinatorial statement about those bundles; one can read the paper's Remark 4.21 as saying the bundle invariant is non-combinatorial exactly beyond that threshold.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The core of this paper is Theorem 2.14, an addition-deletion rule for the minimal degree r(A) of logarithmic derivations: if you delete a hyperplane H and the restriction has r'' larger than r', then r = r' + 1. The proof is short, clean, and uses established tools (Terao's B-theory, Yoshinaga's restriction theorem), with no circularity. This is genuinely new and useful. It yields the Ziegler-extension construction for any d ≥ 9, n3 ≥ 6, and the n3 ≤ 5 classification (Theorem 4.18) is a real result, apart from one SINGULAR computation that is standard and easily checked.

The problem is Theorems 4.10 and 4.11. The displayed equations for A_d and B_d contain factors such as x^{2j} + y^{3j} - z, which are not linear forms. Their zero sets are not lines, so the product does not define a line arrangement. The proofs then say to apply Theorem 3.3 repeatedly and finish by "an elementary counting of intersection points," but that counting only makes sense for line arrangements. As printed, the extremal claims r(A_d) = d-4 and r(B_d) = d-3 are unsupported. These claims are what the abstract advertises as "Tjurina maximal line arrangements in a lot of new situations," so this is a load-bearing gap, not a typo in a footnote.

The core theorem survives, so the paper is repairable. A corrected version needs explicit linear equations for the arrangements and a complete count, or the extremal claims should be removed from the abstract. There is also an indexing typo in the definition of B_{2p+1} ("x^{2d}" rather than a j-dependent exponent), which suggests that section was not checked carefully.

Who is this for? Arrangement and singularity theorists. The main addition-deletion result is a solid contribution and I would cite it. The Tjurina-maximality examples need verification before I would trust them.

Recommendation: send to a serious referee, conditional on the authors fixing or removing the extremal constructions. The paper deserves referee time because the central tool and the n3 ≤ 5 classification are meaningful. I would not desk-reject it, but I would make the fix a condition of acceptance.

Best, [Name]

Referee Report

2 major / 2 minor

Summary. The paper studies the invariant r(A), the minimal degree of a non-Eulerian logarithmic derivation, under the addition or deletion of a hyperplane. The main theorem (Theorem 2.14) gives r(A)=r(A\{H})+1 whenever r(A\{H}) < r(A^H). Section 3 specializes the result to line arrangements, deriving addition and deletion criteria and a weak-combinatorial determination of r when 2m≥d. Section 4 applies these tools to maximal Tjurina line arrangements: explicit families in Theorem 4.4 and Proposition 4.6, extremal families A_d and B_d in Theorems 4.10 and 4.11, and a sharp statement (Theorem 4.18 and Corollary 4.20) that for line arrangements with only double and triple points, r is combinatorially determined exactly when n3≤5, with Ziegler-type counterexamples for every n3≥6.

Significance. The addition-deletion theorem is a genuinely useful tool: its proof is short, clean, and relies on established ingredients such as Terao's polynomial B-theory and Lemma 2.13 rather than assuming its own conclusion. If the Section 4 applications are valid, the paper would establish many new maximal Tjurina line arrangements and a sharp combinatorial-determinacy result for double/triple line arrangements. The explicit grid constructions in Theorem 4.4 are a particular strength. However, the extremal applications in Theorems 4.10 and 4.11 currently depend on displayed equations that are not products of linear forms, so those specific claims are not yet supported. The results on n3≤5 and the Ziegler-based non-determinacy for n3≥6 appear independent of this defect.

major comments (2)

[4.1, displayed equations before Theorem 4.10] The objects A_{3p+2}, A_{3p+3}, and A_{3p+4} are defined by products containing factors such as x^{2j}+y^{3j}-z, x^{2j}+y^{3j+1}-z, and x^{2j}+y^{3j+2}-z. For every j≥1 each such factor has degree at least 3, so its zero set is not a hyperplane; consequently the product is not the defining polynomial of a line arrangement, and the integer d used to name A_d is not the degree of the product. The same problem occurs in the formulas for B_{2p} and B_{2p+1} before Theorem 4.11. The proofs of Theorems 4.10 and 4.11 invoke Theorem 3.3 and the maximal-Tjurina criterion from [18], both of which are statements about line arrangements. As printed, the conclusions r(A_d)=d-4 and r(B_d)=d-3 are unsupported. Correct linear defining equations, or a different proof for the corresponding curves, must be supplied.
[4.1, proof of Theorem 4.10 (and Theorem 4.11)] Even if the defining equations are repaired, the proof of Theorem 4.10 is only a sketch: it states that Theorem 3.3 is applied repeatedly in three addition steps and that an elementary counting of intersection points completes the proof, without giving the restriction data |A^H|, the values of r' and r'', or the counts at any intermediate step. Theorem 4.11 is dismissed by the phrase 'by the same proof'. These counts are load-bearing because the extremal values r=d-4 and r=d-3 are sensitive to the exact multiplicities and to the condition |A^H|≥r'+2 at every intermediate arrangement. The counting must be written out fully, or replaced by a verification covering all d in the claimed ranges, before Theorems 4.10 and 4.11 can be accepted.

minor comments (2)

[4.1, definition of B_{2p+1}] The displayed formula for B_{2p+1} uses an unsubscripted d in the factor x^{2d}+y^{3d}-z and in the product range j=1,...,d-1, while the family is indexed by p; this is internally inconsistent and should be restated with matching indices.
[Abstract and title] There are spacing errors in the title and abstract ('Deriv ations', 'T jurina'); please proofread the manuscript carefully.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the addition-deletion theorem is derived from Terao's and Yoshinaga's ingredients, and the r(A) applications are not fitted. A separate correctness gap affects the A_d and B_d constructions.

full rationale

The main theorem (Theorem 2.14) is proved from Lemma 2.2 and Lemma 2.13, not from its conclusion; the condition r' < r'' is used in a contradiction argument that assumes r = r' and derives a derivation of degree r' - 1 in D0(A'), contradicting minimality. The line-arrangement versions (Theorems 3.3-3.4) are direct applications of Theorem 2.14 plus the fact exp(A^H) = (1, |A^H| - 1), so they inherit this independent footing. The Tjurina-maximal results in Section 4 are based on the external du Plessis-Wall bound (Theorem 4.2) and the combinatorially determined Tjurina formula (1.3), with r computed by the addition theorem, so no parameter is fitted to produce the target r. Theorem 4.18 and Corollary 4.20 similarly use the deletion theorem and Ziegler's published example, not circular reduction. The only self-citation of note is [18] (Dimca-Sticlaru), used for the small-degree base cases and for the equivalence 'maximal Tjurina iff r equals the conjectured value' in Theorems 4.10-4.11; this is a prior result rather than an input assumption of the main derivation, so it is not a circularity, though it is a load-bearing citation in the applications. Separately, the defining equations in Section 4.1 for A_{3p+2} and B_{2p} involve factors such as x^{2j} + y^{3j} - z, which are not linear forms, so the zero sets are not line arrangements; this is a correctness gap in Theorems 4.10-4.11, not a circularity. Overall the derivation chain is self-contained on the central r(A) theorem; score 1.0 reflects the self-citations in the application section rather than any reduction of a prediction to its inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

This paper introduces no free parameters and no invented entities; its derivations are parameter-free. The listed axioms are established theorems or explicit geometric assumptions. The fragile part is the assumption that the printed equations for A_d and B_d define line arrangements, since those equations contain visibly non-linear factors.

assumptions (5)

standard math Terao's polynomial B-theory (Theorem 2.3) and multiple deletion theorem (Theorem 2.4)
Used to compare D(A') and D(A) in Proposition 2.12 and Theorem 2.17; imported as established results.
standard math Yoshinaga's freeness criteria via Ziegler restriction (Theorems 2.5 and 2.6)
Used in Theorem 2.17 and in the restriction arguments in (2.16); prior published theorems.
standard math du Plessis-Wall upper bound for the global Tjurina number (Theorem 4.2)
Defines what it means for an arrangement to be Tjurina maximal; imported from [19,20].
domain assumption The displayed equations in Theorems 4.4, 4.10, and 4.11 define real line arrangements with the asserted intersection counts
The constructions rely on counting |A'∩H| and on each factor being a line; as printed, the factors in 4.10 and 4.11 contain non-linear terms.
domain assumption The SINGULAR computation giving r(B)=3 for the seven-line arrangement B in Theorem 4.18(4B) is correct
The classification for n3=5 case (B) depends on this computation; no code or transcript is provided.

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Cite this review

Pith. "Pith review of Addition-deletion results for the minimal degree of logarithmic derivations of arrangements." pith.science (2026). https://pith.science/paper/CJFDZKT2

@misc{pith2026190806885,
  author       = {Pith},
  title        = {Pith review of: Addition-deletion results for the minimal degree of logarithmic derivations of arrangements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CJFDZKT2}},
  note         = {Machine review of arXiv:1908.06885}
}

abstract

We study the change of the minimal degree of a logarithmic derivation of a hyperplane arrangement under the addition or the deletion of a hyperplane, and give a number of applications. First, we prove the existence of Tjurina maximal line arrangements in a lot of new situations. Then, starting with Ziegler's example of a pair of arrangements of $d=9$ lines with $n_3=6$ triple points in addition to some double points, having the same combinatorics, but distinct minimal degree of a logarithmic derivation, we construct new examples of such pairs, for any number $d\geq 9$ of lines, and any number $n_3\geq 6$ of triple points. Moreover, we show that such examples are not possible for line arrangements having only double and triple points, with $n_3 \leq 5$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

30 extracted references · 28 canonical work pages

[18]

Jacobian syzygies and plane curves with maximal global Tjurina numbers

A. Dimca and G. Sticlaru, Jacobian syzygies and plane curves with maximal global Tjurina numbers. arXiv:1901.05915. 1, 1, 4.1, 4.1, 4.1, 4.12

work page Pith review arXiv 1901
[1]

Abe, Roots of characteristic polynomials and and intersection points of line arrangements

T. Abe, Roots of characteristic polynomials and and intersection points of line arrangements. J. Singularities , 8 (2014), 100–117. 3.5

work page 2014
[2]

Abe, Divisionally free arrangements of hyperplanes

T. Abe, Divisionally free arrangements of hyperplanes. Invent. Math. 204 (2016), no. 1, 317–

work page 2016
[3]

Abe, Deletion theorem and combinatorics of hyperplane arran gements

T. Abe, Deletion theorem and combinatorics of hyperplane arran gements. Math. Ann. 373 (2019), issue 1-2, 581–595. 1

work page 2019
[4]

Abe, Plus-one generated and next to free arrangements of hyperplanes, Int

T. Abe, Plus-one generated and next to free arrangements of hyperplanes, Int. Math. Res. Not.. DOI:10.1093/imrn/rnz099 2.1, 2.7, 2.8, 2.9, 2.10

doi:10.1093/imrn/rnz099
[5]

Abe and A

T. Abe and A. Dimca, On the splitting types of bundles of logarithmic vector ﬁelds along plane curves. Int. J. Math ., 29, no. 8, 1850055, 20 pp. 1, 1, 4.21

work page
[6]

Abe and H

T. Abe and H. Terao, Multiple addition, deletion and restriction the orems for hyperplane arrangements. Proc. Amer. Math. Soc. . DOI:10.1090/proc/14592 2.4

work page doi:10.1090/proc/14592
[7]

D. Cook, B. Harbourne, J. Migliore, U. Nagel, Line arrangements and conﬁgurations of points with an unexpected geometric property. Compositio Math. 154(2018), 2150–2194. 1

work page 2018

Show all 30 references

[8]

Dimca, Hyperplane Arrangements: An Introduction , Universitext, Springer, 2017 1, 3.9

A. Dimca, Hyperplane Arrangements: An Introduction , Universitext, Springer, 2017 1, 3.9

2017
[9]

Dimca, Curve arrangements, pencils, and Jacobian syzygies, Michigan Math

A. Dimca, Curve arrangements, pencils, and Jacobian syzygies, Michigan Math. J. 66 (2017), 347–365. 1, 3.7, 4.13

2017
[10]

Dimca, Freeness versus maximal global Tjurina number for p lane curves, Math

A. Dimca, Freeness versus maximal global Tjurina number for p lane curves, Math. Proc. Cambridge Phil. Soc. 163 (2017), 161–172. 3.7

2017
[11]

Dimca, On rational cuspidal plane curves, and the local coho mology of Jacobian rings, arXiv:1707.05258

A. Dimca, On rational cuspidal plane curves, and the local coho mology of Jacobian rings, arXiv:1707.05258. to appear in Commentarii Mathematici Helvetici. 1

arXiv
[12]

Dimca, D

A. Dimca, D. Ibadula, A. M˘ acinic, Freeness for 13 lines arrange ments is combinatorial, Discrete Mathematics 342 (2019), 2445–2453. 1

2019
[13]

Dimca, G

A. Dimca, G. Sticlaru, Koszul complexes and pole order ﬁltration s, Proc. Edinburg. Math. Soc. 58(2015), 333–354. 4.17 22 TAKURO ABE, ALEXANDRU DIMCA, AND GABRIEL STICLARU

2015
[14]

Dimca, G

A. Dimca, G. Sticlaru, On the exponents of free and nearly free projective plane curves, Rev. Mat. Complut. 30(2017), 259–268. 1, 1

2017
[15]

Dimca, G

A. Dimca, G. Sticlaru, Line and rational curve arrangements, a nd Walther’s inequality, arXiv:1803.05386, to appear in Atti Accad. Naz. Lincei Rend. Lince i Mat. Appl. 4.21

arXiv
[16]

Dimca, G

A. Dimca, G. Sticlaru, On the jumping lines of bundles of logarithmic vector ﬁelds along plane curves, arXiv: 1804.06349

arXiv
[17]

Dimca, G

A. Dimca, G. Sticlaru, Plane curves with three syzygies, minimal T jurina curves, and nearly cuspidal curves, arXiv: 1810.11766. 1

arXiv
[19]

du Plessis, C.T.C

A.A. du Plessis, C.T.C. Wall, Application of the theory of the discrimin ant to highly singular plane curves, Math. Proc. Camb. Phil. Soc. , 126 (1999), 259-266. 3.7, 4.1

1999
[20]

Ellia, Quasi complete intersections and global Tjurina numbe r of plane curves, J

Ph. Ellia, Quasi complete intersections and global Tjurina numbe r of plane curves, J. Pure Appl. Algebra (2019), https://doi.org/10.1016/j.jpaa.2019.05.014 . 4.1

2019 doi
[21]

Marchesi, J

S. Marchesi, J. Vall` es, Terao’s conjecture for triangular ar rangements, arXiv:1903.08885. 1, 1

1903 arXiv
[22]

Orlik and H

P. Orlik and H. Terao, Arrangements of Hyperplanes, Springer-Verlag, Berlin Heidelberg New York, 1992. 1

1992
[23]

Schenck, H

H. Schenck, H. Terao, M. Yoshinaga, Logarithmic vector ﬁelds for curve conﬁgurations in P2 with quasihomogeneous singularities, Math. Res. Lett. 25 (2018), 1977–1992. 1

2018
[24]

Simis, S.O

A. Simis, S.O. Toh˘ aneanu, Homology of homogeneous divisors, I srael J. Math. 200 (2014), 449-487

2014
[25]

Terao, Arrangements of hyperplanes and their freeness I , II

H. Terao, Arrangements of hyperplanes and their freeness I , II. J. Fac. Sci. Univ. Tokyo 27 (1980), 293–320. 2.3, 3.2

1980
[26]

Wakeﬁeld, Derivation degree sequences of non-free arran gements, Communications in Al- gebra, 47:7(2019), 2654–2666, DOI: 10.1080/00927872.2018.15 36207

M. Wakeﬁeld, Derivation degree sequences of non-free arran gements, Communications in Al- gebra, 47:7(2019), 2654–2666, DOI: 10.1080/00927872.2018.15 36207. 1

2019
[27]

Yoshinaga, On the freeness of 3-arrangements

M. Yoshinaga, On the freeness of 3-arrangements. Bull. London Math. Soc. 37 (2005), no. 1, 126–134. 2.6

2005
[28]

Yoshinaga, Freeness of hyperplane arrangements and rela ted topics

M. Yoshinaga, Freeness of hyperplane arrangements and rela ted topics. Annales de la Faculte des Sciences de Toulouse , 23 (2014), no. 2, 483–512. 1, 2.2

2014
[29]

G. M. Ziegler, Multiarrangements of hyperplanes and their free ness. Singularities (Iowa City, IA, 1986), 345–359, 2.5

1986
[30]

Ziegler, Combinatorial construction of logarithmic diﬀerentia l forms, Adv

G. Ziegler, Combinatorial construction of logarithmic diﬀerentia l forms, Adv. Math. 76 (1989), 116-154. 3.9 Institute of Mathematics for Industry, Kyushu University, Fukuoka 819-0395, Japan. E-mail address : abe@imi.kyushu-u.ac.jp Universit´ e C ˆote d’Azur, CNRS, LJAD, Franc...

1989

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [18]

Jacobian syzygies and plane curves with maximal global Tjurina numbers

A. Dimca and G. Sticlaru, Jacobian syzygies and plane curves with maximal global Tjurina numbers. arXiv:1901.05915. 1, 1, 4.1, 4.1, 4.1, 4.12

work page Pith review arXiv 1901

[2] [1]

Abe, Roots of characteristic polynomials and and intersection points of line arrangements

T. Abe, Roots of characteristic polynomials and and intersection points of line arrangements. J. Singularities , 8 (2014), 100–117. 3.5

work page 2014

[3] [2]

Abe, Divisionally free arrangements of hyperplanes

T. Abe, Divisionally free arrangements of hyperplanes. Invent. Math. 204 (2016), no. 1, 317–

work page 2016

[4] [3]

Abe, Deletion theorem and combinatorics of hyperplane arran gements

T. Abe, Deletion theorem and combinatorics of hyperplane arran gements. Math. Ann. 373 (2019), issue 1-2, 581–595. 1

work page 2019

[5] [4]

Abe, Plus-one generated and next to free arrangements of hyperplanes, Int

T. Abe, Plus-one generated and next to free arrangements of hyperplanes, Int. Math. Res. Not.. DOI:10.1093/imrn/rnz099 2.1, 2.7, 2.8, 2.9, 2.10

doi:10.1093/imrn/rnz099

[6] [5]

Abe and A

T. Abe and A. Dimca, On the splitting types of bundles of logarithmic vector ﬁelds along plane curves. Int. J. Math ., 29, no. 8, 1850055, 20 pp. 1, 1, 4.21

work page

[7] [6]

Abe and H

T. Abe and H. Terao, Multiple addition, deletion and restriction the orems for hyperplane arrangements. Proc. Amer. Math. Soc. . DOI:10.1090/proc/14592 2.4

work page doi:10.1090/proc/14592

[8] [7]

D. Cook, B. Harbourne, J. Migliore, U. Nagel, Line arrangements and conﬁgurations of points with an unexpected geometric property. Compositio Math. 154(2018), 2150–2194. 1

work page 2018