In this note we study the logarithmic derivation module of a non-free arrangement. We prove a generalized addition theorem for all arrangements. This addition theorem allows us to find various relationships between non-free arrangements, free arrangements and restriction counts. For graphic arrangements we can use these results to find a lower bound for the maximal degree generator in terms of triangles in the associated graph. We also apply these results to the case of hypersolvable arrangements where we define hyperexponents and use them to find a lower bound for their maximal degree generator.
In these notes we study hyperplane arrangements having at least one logarithmic derivation of degree two that is not a combination of degree one logarithmic derivations. It is well-known that if a hyperplane arrangement has a linear logarithmic derivation not a constant multiple of the Euler derivation, then the arrangement decomposes as the direct product of smaller arrangements. The next natural step would be to study arrangements with non-trivial quadratic logarithmic derivations. On this regard, we present a computational lemma that leads to a full classification of hyperplane arrangements…
The separator-based derivations for arrangements of monomial groups
Center for University Mathematics Teaching, College of Science, Nanjing Forestry University（南京林业大学理学院）；Department of Mathematics, College of Mathematics and Physics, Beijing University of Chemical Technology（北京化工大学数理学院）
Let A be an n by d matrix having full rank n. An orthogonal dual A^{\perp} of A is a (d-n) by d matrix of rank (d-n) such that every row of A^{\perp} is orthogonal (under the usual dot product) to every row of A. We define the orthogonal dual for arrangements by identifying an essential (central) arrangement of d hyperplanes in n-dimensional space with the n by d matrix of coefficients of the homogeneous linear forms for which the hyperplanes are kernels. If n is at least 5, we show that if the matroid (or the intersection lattice) of an n-dimensional essential arrangement A contains a modular…
We study the Hadamard product of the linear forms defining a hyperplane arrangement with those of its dual, which we view as generating an ideal in a certain polynomial ring. We use this ideal, which we call the ideal of pairs, to study logarithmic derivations and critical set varieties of arrangements in a way which is symmetric with respect to matroid duality. Our main result exhibits the variety of the ideal of pairs as a subspace arrangement whose components correspond to cyclic flats of the arrangement. As a corollary, we are able to give geometric explanations of some freeness and projec…
Let $\A$ be an irreducible Coxeter arrangement and $\bfk$ be a multiplicity of $\A$. We study the derivation module $D(\A, \bfk)$. Any two-dimensional irreducible Coxeter arrangement with even number of lines is decomposed into two orbits under the action of the Coxeter group. In this paper, we will {explicitly} construct a basis for $D(\A, \bfk)$ assuming $\bfk$ is constant on each orbit. Consequently we will determine the exponents of $(\A, \bfk)$ under this assumption. For this purpose we develop a theory of universal derivations and introduce a map to deal with our exceptional cases.
We will prove the freeness of multi-Coxeter arrangements by constructing a basis of the module of vector fields which contact to each reflecting hyperplanes with some multiplicities using K. Saito's theory of primitive derivation.
We define {\bf primitive derivations} for Coxeter arrangements which may not be irreducible. Using those derivations, we introduce the {\bf primitive filtrations} of the module of invariant logarithmic differential forms for an arbitrary Coxeter arrangement with an arbitrary multiplicity. In particular, when the Coxeter arrangement is irreducible with a constant multiplicity, the primitive filtration has already been studied, which generalizes the Hodge filtration introduced by K. Saito.
Let $W$ be a finite irreducible real reflection group, which is a Coxeter group. We explicitly construct a basis for the module of differential 1-forms with logarithmic poles along the Coxeter arrangement by using a primitive derivation. As a consequence, we extend the Hodge filtration, indexed by nonnegative integers, into a filtration indexed by all integers. This filtration coincides with the filtration by the order of poles. The results are translated into the derivation case.
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 …
In [5] (=Terao, H.: Multiderivations of Coxeter arrangements. Inventiones math., 148 (2002) 659--674), we constructed a basis for the contact-order filtration of the module of derivations on the orbit space of a finite real reflection group acting on an $\ell$-dimensional Euclidean space. Recently M. Yoshinaga constructed another basis for the contact-order filtration in [7] (=Yoshinaga, M.: The primitive derivation and freeness of multi-Coxeter arrangements. preprint 2002). In this note we give an explicit formula relating Yoshinaga's basis to the basis given in [5]. The two bases turn out to…
The Hodge filtration of the module of derivations on the orbit space of a finite real reflection group acting on an $\ell$-dimensional Euclidean space was introduced and studied by K. Saito. The filtration is equivalent data to the flat structure or the Frobenius manifold structure. We will show that the Hodge filtration coincides with the filtration by the order of contacts to the reflecting hyperplanes. Moreover, a standard basis for the Hodge filtration is explicitly given.
The modules of logarithmic derivations for the (extended) Catalan and Shi arrangements associated with root systems are known to be free. However, except for a few cases, explicit bases for such modules are not known. In this paper, we construct explicit bases for type $A$ root systems. Our construction is based on Bandlow-Musiker's integral formula for a basis of the space of quasiinvariants. The integral formula can be considered as an expression for the inverse of the primitive derivation introduced by K. Saito. We prove that the discrete analogues of the integral formulas provide bases for…
In his affirmative answer to the Edelman-Reiner conjecture, Yoshinaga proved that the logarithmic derivation modules of the cones of the extended Shi arrangements are free modules. However, all we know about the bases is their existence and degrees. In this article, we introduce two distinguished bases for the modules. More specifically, we will define and study the simple-root basis plus (SRB+) and the simple-root basis minus (SRB-) when a primitive derivation is fixed. They have remarkable properties relevant to the simple roots and those properties characterize the bases.
A pseudoline arrangement graph is a planar graph induced by an embedding of a (simple) pseudoline arrangement. We study the corresponding graph realization problem and properties of pseudoline arrangement graphs. In the first part, we give a simple criterion based on the degree sequence that says whether a degree sequence will have a pseudoline arrangement graph as one of its realizations. In the second part, we study the eccentricities of vertices in such graphs. We observe that the diameter (maximum eccentricity of a vertex in the graph) of any pseudoline arrangement graph on $n$ pseudolines…
Let $\A$ be an irreducible Coxeter arrangement and $W$ be its Coxeter group. Then $W$ naturally acts on $\A$. A multiplicity $\bfm : \A\rightarrow \Z$ is said to be equivariant when $\bfm$ is constant on each $W$-orbit of $\A$. In this article, we prove that the multi-derivation module $D(\A, \bfm)$ is a free module whenever $\bfm$ is equivariant by explicitly constructing a basis, which generalizes the main theorem of \cite{T02}. The main tool is a primitive derivation and its covariant derivative. Moreover, we show that the $W$-invariant part $D(\A, \bfm)^{W}$ for any multiplicity $\bfm$ is …
We study curves consisting of unions of projective lines whose intersections are given by graphs. Under suitable hypotheses on the graph, these so-called \emph{graph curves} can be embedded in projective space as line arrangements. We discuss property $N_p$ for these embeddings and are able to produce products of linear forms that generate the ideal in certain cases. We also briefly discuss questions regarding the higher-dimensional subspace arrangements obtained by taking the secant varieties of graph curves.
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…
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.
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.
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…
We prove Terao conjecture saying that the freeness is determined by the combinatorics for arrangements of 13 lines in the complex projective plane and that the property of being nearly free is combinatorial for line arrangements of up to 12 lines in the complex projective plane.
We show that the ideal of an arrangement of d linear subspaces of projective space is d-regular in the sense of Castelnuovo and Mumford, answering a question of B. Sturmfels. In particular this implies that the ideal of an arrangement of d subspaces is generated in degrees less than or equal to d.
Let $V$ be an $\ell$-dimensional Euclidean space. Let $G \subset O(V)$ be a finite irreducible orthogonal reflection group. Let ${\cal A}$ be the corresponding Coxeter arrangement. Let $S$ be the algebra of polynomial functions on $V.$ For $H \in {\cal A}$ choose $α_H \in V^*$ such that $H = {\rm ker}(α_H).$ For each nonnegative integer $m$, define the derivation module $\sD^{(m)}({\cal A}) = \{θ\in {\rm Der}_S | θ(α_H) \in S α^m_H\}$. The module is known to be a free $S$-module of rank $\ell$ by K. Saito (1975) for $m=1$ and L. Solomon-H. Terao (1998) for $m=2$. The main result of this paper …
We show that points in specific degree 2 hypersurfaces in the Grassmannian $Gr(3, n)$ correspond to generic arrangements of $n$ hyperplanes in $\mathbb{C}^3$ with associated discriminantal arrangement having intersections of multiplicity three in codimension two.
Any stretching of Ringel's non-Pappus pseudoline arrangement when projected into the Euclidean plane, implicitly contains a particular arrangement of nine triangles. This arrangement has a complex constraint involving the sines of its angles. These constraints cannot be satisfied by any projection of the initial arrangement. This is sufficient to prove Pappus's theorem. The derivation of the constraint is via systems of inequalities arising from the polar coordinates of the lines. These systems are linear in r for any given theta, and their solubility can be analysed in terms of the signs of d…
Let $X$ be a nonempty real variety that is invariant under the action of a reflection group $G$. We conjecture that if $X$ is defined in terms of the first $k$ basic invariants of $G$ (ordered by degree), then $X$ meets a $k$-dimensional flat of the associated reflection arrangement. We prove this conjecture for the infinite types, reflection groups of rank at most $3$, and $F_4$ and we give computational evidence for $H_4$. This is a generalization of Timofte's degree principle to reflection groups. For general reflection groups, we compute nontrivial upper bounds on the minimal dimension of …
Free line arrangements with low maximal multiplicity
Université Côte d’Azur（蔚蓝海岸大学）；Simion Stoilow Institute of Mathematics（西蒙·斯托伊洛瓦数学研究所）；Universität Bielefeld（比勒费尔德大学）；University of the National Education Commission Krakow（克拉科夫国民教育委员会大学）
We obtain a new bound on the number of two-rich points spanned by an arrangement of low degree algebraic curves in $\mathbb{R}^4$. Specifically, we show that an arrangement of $n$ algebraic curves determines at most $C_εn^{4/3+3ε}$ two-rich points, provided at most $n^{2/3+2ε}$ curves lie in any low degree hypersurface and at most $n^{1/3+ε}$ curves lie in any low degree surface. This result follows from a structure theorem about arrangements of curves that determine many two-rich points.
We prove the torsion freeness of the decomposable Orlik--Solomon algebra of a simple matroid on ground set $[n]$. In the class of hypersolvable \& non-supersolvable complex hyperplane arrangements, the torsion freeness, in a certain degree, of this combinatorially defined object, associated to the intersection lattice of the arrangement, impacts on the first non-vanishing higher homotopy group of the complement of the arrangement.
We show that the maximum number of pairwise non-overlapping $k$-rich lenses (lenses formed by at least $k$ circles) in an arrangement of $n$ circles in the plane is $O\left(\frac{n^{3/2}\log{(n/k^3)}}{k^{5/2}} + \frac{n}{k} \right)$, and the sum of the degrees of the lenses of such a family (where the degree of a lens is the number of circles that form it) is $O\left(\frac{n^{3/2}\log{(n/k^3)}}{k^{3/2}} + n\right)$. Two independent proofs of these bounds are given, each interesting in its own right (so we believe). We then show that these bounds lead to the known bound of Agarwal et al. (JACM …
We show that the maximum likelihood degree of a smooth very affine variety is equal to the signed topological Euler characteristic. This generalizes Orlik and Terao's solution to Varchenko's conjecture on complements of hyperplane arrangements to smooth very affine varieties. For very affine varieties satisfying a genericity condition at infinity, the result is further strengthened to relate the variety of critical points to the Chern-Schwartz-MacPherson class. The strengthened version recovers the geometric deletion-restriction formula of Denham et al. for arrangement complements, and general…
We show that a set of $n$ algebraic plane curves of constant maximum degree can be cut into $O(n^{3/2}\operatorname{polylog} n)$ Jordan arcs, so that each pair of arcs intersect at most once, i.e., they form a collection of pseudo-segments. This extends a similar (and slightly better) bound for pseudo-circles due to Marcus and Tardos. Our result is based on a technique of Ellenberg, Solymosi and Zahl that transforms arrangements of plane curves into arrangements of space curves, so that lenses (pairs of subarcs of the curves that intersect at least twice) become vertical depth cycles. We then …
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. Marche…
We examine the first non-vanishing higher homotopy group, $π_p$, of the complement of a hypersolvable, non--supersolvable, complex hyperplane arrangement, as a module over the group ring of the fundamental group, $\Zπ_1$. We give a presentation for the $I$--adic completion of $π_p$. We deduce that the second nilpotent $I$--adic quotient of $π_p$ is determined by the combinatorics of the arrangement, and we give a combinatorial formula for the second associated graded piece, $\gr^1_I π_p$. We relate the torsion of this graded piece to the dimensions of the minimal generating systems of the Orli…
Let \A be a complex hyperplane arrangement, with fundamental group G and holonomy Lie algebra \H. Suppose \H_3 is a free abelian group of minimum possible rank, given the values the Möbius function μ: Ł_2\to \Z takes on the rank 2 flats of \A. Then the associated graded Lie algebra of G decomposes (in degrees 2 and higher) as a direct product of free Lie algebras. In particular, the ranks of the lower central series quotients of the group are given by ϕ_r(G)=\sum_{X\in Ł_2} ϕ_r(F_{μ(X)}), for r\ge 2. We illustrate this new Lower Central Series formula with several families of examples.
If $M$ is the complement of a hyperplane arrangement, and $A=H^*(M,\k)$ is the cohomology ring of $M$ over a field of characteristic 0, then the ranks, $ϕ_k$, of the lower central series quotients of $π_1(M)$ can be computed from the Betti numbers, $b_{ii}=\dim_{\k} \Tor^A_i(\k,\k)_i$, of the linear strand in a (minimal) free resolution of $\k$ over $A$. We use the Cartan-Eilenberg change of rings spectral sequence to relate these numbers to the graded Betti numbers, $b'_{ij}=\dim_{\k} \Tor^E_i(A,\k)_j$, of a (minimal) resolution of $A$ over the exterior algebra $E$. From this analysis, we rec…
