arXiv subjects

Takuro Abe

Publications and source records attributed to Takuro Abe.

At least 37 records · Page 2Linked to original sources

arXiv2022

Generalization of the addition and restriction theorems from free arrangements to the class of projective dimension one

We study a generalized version of Terao's famous addition theorem for free arrangements to the category of those with projective dimension one. Namely, we give a criterion to determine the algebraic structure of logarithmic derivation modules of the addition when the deletion and restrictions are free with a mild condition. Also, we introduce a class of divisionally SPOG arrangements whose SPOGness depends only on the intersection lattice like Terao's famous conjecture on combinatoriality of freeness.

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arXiv2021

Vertex-weighted Digraphs and Freeness of Arrangements Between Shi and Ish

We introduce and study a digraph analogue of Stanley's $ψ$-graphical arrangements from the perspectives of combinatorics and freeness. Our arrangements form a common generalization of various classes of arrangements in literature including the Catalan arrangement, the Shi arrangement, the Ish arrangement, and especially the arrangements interpolating between Shi and Ish recently introduced by Duarte and Guedes de Oliveira. The arrangements between Shi and Ish all are proved to have the same characteristic polynomial with all integer roots, thus raising the natural question of their freeness. We define two operations on digraphs, which we shall call king and coking elimination operations and prove that subject to certain conditions on the weight $ψ$, the operations preserve the characteristic polynomials and freeness of the associated arrangements. As an application, we affirmatively prove that the arrangements between Shi and Ish all are free, and among them only the Ish arrangement has supersolvable cone.

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arXiv2021

Projective dimensions of hyperplane arrangements

We establish a general theory for projective dimensions of the logarithmic derivation modules of hyperplane arrangements. That includes the addition-deletion and restriction theorem, Yoshinaga-type result, and the division theorem for projective dimensions of hyperplane arrangements. They are generalizations of the free arrangement cases, that can be regarded as the special case of our result when the projective dimension is zero. The keys to prove them are several new methods to determine the surjectivity of the Euler and the Ziegler restriction maps, that is combinatorial when the projective dimension is not maximal for all localizations. Also, we introduce a new class of arrangements in which the projective dimension is comibinatorially determined.

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arXiv2021

Roots of the characteristic polynomials of hyperplane arrangements and their restrictions and localizations

Terao's factorization theorem shows that if an arrangement is free, then its characteristic polynomial factors into the product of linear polynomials over the integer ring. This is not a necessary condition, but there are not so many non-free arrangements whose characteristic polynomial factors over the integer ring. On the other hand, the localization of a free arrangement is free, and its restriction is in many cases free, thus its characteristic polynomial factors. In this paper, we consider how their integer, or real roots behave.

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arXiv2021

Locally Heavy Hyperplanes in Multiarrangements

Hyperplane Arrangements of rank $3$ admitting an unbalanced Ziegler restriction are known to fulfill Terao's conjecture. This long-standing conjecture asks whether the freeness of an arrangement is determined by its combinatorics. In this note, we prove that arrangements that admit a locally heavy flag satisfy Terao's conjecture which is a generalization of the statement above to arbitrary dimension. To this end, we extend results characterizing the freeness of multiarrangements with a heavy hyperplane to those satisfying the weaker notion of a locally heavy hyperplane. As a corollary, we give a new proof that irreducible arrangements with a generic hyperplane are totally non-free. In another application, we show that an irreducible multiarrangement of rank $3$ with at least two locally heavy hyperplanes is not free.

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arXiv2020

On $A_1^2$ restrictions of Weyl arrangements

Let $\mathcal{A}$ be a Weyl arrangement in an $\ell$-dimensional Euclidean space. The freeness of restrictions of $\mathcal{A}$ was first settled by a case-by-case method by Orlik and the second author (1993), and later by a uniform argument by Douglass (1999). Prior to this, Orlik and Solomon (1983) had completely determined the exponents of these arrangements by exhaustion. A classical result due to Orlik, Solomon and the second author (1986), asserts that the exponents of any $A_1$ restriction, i.e., the restriction of $\mathcal{A}$ to a hyperplane, are given by $\{m_1,\ldots, m_{\ell-1}\}$, where $\exp(\mathcal{A})=\{m_1,\ldots, m_{\ell}\}$ with $m_1 \le \cdots\le m_{\ell}$. As a next step towards conceptual understanding of the restriction exponents we will investigate the $A_1^2$ restrictions, i.e., the restrictions of $\mathcal{A}$ to the subspaces of type $A_1^2$. In this paper, we give a combinatorial description of the exponents and describe bases for the modules of derivations of the $A_1^2$ restrictions in terms of the classical notion of related roots by Kostant (1955).

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arXiv2019

Double points of free projective line arrangements

We prove Anzis and Tohaneanu conjecture, that is the Dirac-Motzkin conjecture for supersolvable line arrangements in the projective plane over an arbitrary field of characteristic zero. Moreover, we show that a divisionally free arrangements of lines contain at least one double point, that can be regarded as the Sylvester-Gallai theorem for some free arrangements. This is a corollary of a general result that if you add a line to a free projective line arrangement, then that line has to contain at least one double point. Also we prove some conjectures and one open problems related to supersolvable line arrangements and the number of double points.

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arXiv2019

On complex supersolvable line arrangements

We show that the number of lines in an $m$--homogeneous supersolvable line arrangement is upper bounded by $3m-3$ and we classify the $m$--homogeneous supersolvable line arrangements with two modular points up-to lattice-isotopy. A lower bound for the number of double points $n_2$ in an $m$--homogeneous supersolvable line arrangement of $d$ lines is also considered. When $3 \leq m \leq 5$, or when $m \geq \frac{d}{2}$, or when there are at least two modular points, we show that $n_2 \geq \frac{d}{2}$, as conjectured by B. Anzis and S. O. Toh\u aneanu. This conjecture is shown to hold also for supersolvable line arrangements obtained as cones over generic line arrangements, or cones over arbitrary line arrangements having a generic vertex.

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arXiv2019

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

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 $n_3 \leq 5$.

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arXiv2018

Addition-deletion theorem for free hyperplane arrangements and combinatorics

In the theory of hyperplane arrangements, the most important and difficult problem is the combinatorial dependency of several properties. In this atricle, we prove that Terao's celebrated addition-deletion theorem for free arrangements is combinatorial, i.e., whether you can apply it depends only on the intersection lattice of arrangements. The proof is based on a classical technique. Since some parts are already completed recently, we prove the rest part, i.e., the combinatoriality of the addition theorem. As a corollary, we can define a new class of free arrangements called the additionally free arrangement of hyperplanes, which can be constructed from the empty arrangement by using only the addition theorem. Then we can show that Terao's conjecture is true in this class. As an application, we can show that every ideal-Shi arrangement is additionally free, implying that their freeness is combinatorial.

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arXiv2018

Plus-one generated and next to free arrangements of hyperplanes

We introduce a new class of arrangements of hyperplanes, called (strictly) plus-one generated arrangements, from algebraic point of view. Plus-one generatedness is close to freeness, i.e., plus-one generated arrangements have their logarithmic derivation modules generated by dimension plus one elements, with relations containing one linear form coefficient. We show that strictly plus-one generated arrangements can be obtained if we delete a hyperplane from free arrangements. We show a relative freeness criterion in terms of plus-one generatedness. In particular, for plane arrangements, we show that a free arrangement is in fact surrounded by free or strictly plus-one generated arrangements. We also give several applications.

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arXiv2018

Solomon-Terao algebra of hyperplane arrangements

We introduce a new algebra associated with a hyperplane arrangement $\mathcal{A}$, called the Solomon-Terao algebra $\mbox{ST}(\mathcal{A},η)$, where $η$ is a homogeneous polynomial. It is shown by Solomon and Terao that $\mbox{ST}(\mathcal{A},η)$ is Artinian when $η$ is generic. This algebra can be considered as a generalization of coinvariant algebras in the setting of hyperplane arrangements. The class of Solomon-Terao algebras contains cohomology rings of regular nilpotent Hessenberg varieties. We show that $\mbox{ST}(\mathcal{A},η)$ is a complete intersection if and only if $\mathcal{A}$ is free. We also give a factorization formula of the Hilbert polynomials when $\mathcal{A}$ is free, and pose several related questions, problems and conjectures.

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arXiv2018

Multiple addition, deletion and restriction theorems for hyperplane arrangements

In the study of free arrangements, the most useful result to construct/check free arrangements is the addition-deletion theorem. Recently, the multiple version of the addition theorem is proved, called the multiple addition theorem (MAT) to prove the ideal-free theorem. The aim of this article is to give the deletion version of MAT, the multiple deletion theorem (MDT). Also, we can generalize MAT from the viewpoint of our new proof. Moreover, we introduce their restriction version, a multiple restriction theorem (MRT). Applications of them including the combinatorial freeness of the extended Catalan arrangements are given.

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arXiv2017

A characterization of high order freeness for product arrangements and answers to Holm's questions

An m-free hyperplane arrangement is a generalization of a free arrangement. Holm asked the following two questions: (1)Does m-free imply (m+1)-free for any arrangement? (2)Are all arrangements m-free for m large enough? In this paper, we characterize m-freeness for product arrangements, while we prove that all localizations of an m-free arrangement are m-free. From these results, we give answers to Holm's questions.

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arXiv2017

Deletion theorem and combinatorics of hyperplane arrangements

We show that the deletion theorem of a free arrangement is combinatorial, i.e., whether we can delete a hyperplane from a free arrangement keeping freeness depends only on the intersection lattice. In fact, we give an explicit sufficient and necessary condition for the deletion theorem in terms of characteristic polynomials. This gives a lot of corollaries including the existence of free filtrations. The proof is based on the result about the form of minimal generators of a logarithmic derivation module of a multiarrangement which satisfies the $b_2$-equality.

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arXiv2017

Signed-eliminable graphs and free multiplicities on the braid arrangement

We define specific multiplicities on the braid arrangement by using edge-bicolored graphs. To consider their freeness, we introduce the notion of bicolor-eliminable graphs as a generalization of Stanley's classification theory of free graphic arrangements by chordal graphs. This generalization gives us a complete classification of the free multiplicities defined above. As an application, we prove one direction of a conjecture of Athanasiadis on the characterization of the freeness of the deformation of the braid arrangement in terms of directed graphs.

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arXiv2017

On the splitting types of bundles of logarithmic vector fields along plane curves

We give a formula relating the total Tjurina number and the generic splitting type of the bundle of logarithmic vector fields associated to a reduced plane curve. By using it, we give a characterization of nearly free curves in terms of splitting types. Several applications to free and nearly free arrangements of lines are also given, in particular a proof of a form of Terao's Conjecture for arrangements having a line with at most 4 intersection points.

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arXiv2017

Divisionally free arrangements of hyperplanes

We consider the triple $(\mathcal{A},\mathcal{A}',\mathcal{A}^H)$ of hyperplane arrangements and the division of their characteristic polynomials. We show that the freeness of $\mathcal{A}^H$ and the division of $χ(\mathcal{A};t)$ by $χ(\mathcal{A}^H;t)$ confirm the freeness of $\mathcal{A}$. The key ingredient of this "division theorem" on freeness is the fact that, if $χ(\mathcal{A}^H;t)$ divides $χ(\mathcal{A};t)$, then the same holds for the localization at the codimension three flat in $H$. This implies the local-freeness of $\mathcal{A}$ in codimension three along $H$. Based on these results, several applications are obtained, which include a definition of "divisionally free arrangements". It is strictly larger than the set of inductively free arrangements. Also, in the set of divisionally free arrangements, the Terao's conjecture is true.

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