Subspace arrangements defined by products of linear forms
Anders Björner, Irena Peeva, Jessica Sidman
备注：
20 pages; AMSLatex; v.2: proof of Proposition 5.1.3 corrected; proof of Proposition 5.1.6 shortened; references added, v.3: minor corrections; final version; to appear in the Journal of the London Mathematical Society
发表信息：
J. London Math. Soc. (2) 71 (2005), no. 2, 273-288
We consider the vanishing ideal of an arrangement of linear subspaces in a vector space and investigate when this ideal can be generated by products of linear forms. We introduce a combinatorial construction (blocker duality) which yields such generators in cases with a lot of combinatorial structure, and we present the examples that motivated our work. We give a construction which produces all elements of this type in the vanishing ideal of the arrangement. This leads to an algorithm for deciding if the ideal is generated by products of linear forms. We also consider generic arrangements of points in ${\bf P}^2$ and lines in ${\bf P}^3.$
We give explicit generators for ideals of two classes of subspace arrangements embedded in certain reflection arrangements, generalizing results of Li-Li and Kleitman-Lovasz. We also give minimal generators for the ideals of arrangements that arise in a natural way from the p-skeleton of an n-dimensional cube and discuss conditions under which the generators that we give form a Groebner basis.
The vanishing ideal I of a subspace arrangement is an intersection of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of a product J of the linear ideals without any assumptions on the subspace arrangement. It turns out that the Hilbert series of J is a combinatorial invariant of the subspace arrangement: it only depends on the intersection lattice and the dimension function. The graded Betti numbers of J are determined by the Hilbert series, so they are combinatorial invariants as well. The res…
For a subspace arrangement over a finite field we study the evaluation code defined on the arrangement's set of points. The length of this code is given by the subspace arrangement's characteristic polynomial. For coordinate subspace arrangements, the dimension is bounded below by the face vector of the corresponding simplicial complex. The minimum distance is determined for coordinate subspace arrangements where the simplicial complex is a skeleton.
Given a simplicial hyperplane arrangement H and a subspace arrangement A embedded in H, we define a simplicial complex Delta_{A,H} as the subdivision of the link of A induced by H. In particular, this generalizes Steingrimsson's coloring complex of a graph. We do the following: (1) When A is a hyperplane arrangement, Delta_{A,H} is shown to be shellable. As a special case, we answer affirmatively a question of Steingrimsson on coloring complexes. (2) For H being a Coxeter arrangement of type A or B we obtain a close connection between the Hilbert series of the Stanley-Reisner ring of Delta_{A,…
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.
The homotopy type of the complement of a complex coordinate subspace arrangement is studied by fathoming out the connection between its topological and combinatorial structures. A family of arrangements for which the complement is homotopy equivalent to a wedge of spheres is described. One consequence is an application in commutative algebra: certain local rings are proved to be Golod, that is, all Massey products in their homology vanish.
Let $I_1,\dots,I_n$ be ideals generated by linear forms in a polynomial ring over an infinite field and let $J = I_1 \cdots I_n$. We describe a minimal free resolution of $J$ and show that it is supported on a polymatroid obtained from the underlying representable polymatroid by means of the so-called Dilworth truncation. Formulas for the projective dimension and Betti numbers are given in terms of the polymatroid as well as a characterization of the associated primes. Along the way we show that $J$ has linear quotients. In fact, we do this for a large class of ideals $J_P$, where $P$ is a cer…
We show that, for an arrangement of subspaces in a complex vector space with geometric intersection lattice, the complement of the arrangement is formal. We prove that the Morgan rational model for such an arrangement complement is formal as a differential graded algebra.
In this paper, we study $k$-parabolic arrangements, a generalization of $k$-equal arrangements for finite real reflection groups. When $k=2$, these arrangements correspond to the well-studied Coxeter arrangements. Brieskorn (1971) showed that the fundamental group of the complement, over $\mathbb{C}$, of the type $W$ Coxeter arrangement is isomorphic to the pure Artin group of type $W$. Khovanov (1996) gave an algebraic description for the fundamental group of the complement, over $\mathbb{R}$, of the 3-equal arrangement. We generalize Khovanov's result to obtain an algebraic description of th…
We exhibit a family of real rotation groups whose subspace arrangements are not contained in that of any real reflection group, answering a question of Martino and Singh.
We investigate in this paper a generalization of Solomon-Terao formula for central equidimensional subspace arrangements. We introduce generalized Solomon-Terao functions based on the Hilbert-Poincaré series of the modules of multi-logarithmic forms and logarithmic multi-residues. We show that as in the case of hyperplane arrangements, these Solomon-Terao functions are polynomial. We then prove that if the Solomon-Terao polynomial of the modules of multi-residues satisfies a certain property, then this polynomial is related to the characteristic polynomial of the subspace arrangement. In parti…
For each positive integer $n \geq 4$, we give an inequality satisfied by rank functions of arrangements of $n$ subspaces. When $n=4$ we recover Ingleton's inequality; for higher $n$ the inequalities are all new. These inequalities can be thought of as a hierarchy of necessary conditions for a (poly)matroid to be realizable. Some related open questions about the "cone of realizable polymatroids" are also presented.
In this paper we represent the Vassiliev model for the homotopy type of the one-point compactification of subspace arrangements as a homotopy colimit of an appropriate diagram over the nerve complex of the intersection semilattice of the arrangement. Furthermore, using a generalization of simplicial collapses to diagrams of topological spaces over simplicial complexes, we construct an explicit deformation retraction from the Vassiliev model to the Ziegler-Zivaljevic model.
In this paper, we study $k$-parabolic arrangements, a generalization of the $k$-equal arrangement for any finite real reflection group. When $k=2$, these arrangements correspond to the well-studied Coxeter arrangements. We construct a cell complex $Perm_k(W)$ that is homotopy equivalent to the complement. We then apply discrete Morse theory to obtain a minimal cell complex for the complement. As a result, we give combinatorial interpretations for the Betti numbers, and show that the homology groups are torsion free. We also study a generalization of the Independence Complex of a graph, and sho…
We give a combinatorial characterization of isotropic subspaces in the Orlik- Solomon algebra of a hyperplane arrangement in terms of decorations of its intersection lattice. We then use this characterization to prove a result that relates these isotropic subspaces with linear systems supported on the arrangement, for arrangements with isolated non-normal crossings of a particular form.
In this work we study arrangements of $k$-dimensional subspaces $V_1,\ldots,V_n \subset \mathbb{C}^\ell$. Our main result shows that, if every pair $V_{a},V_b$ of subspaces is contained in a dependent triple (a triple $V_{a},V_b,V_c$ contained in a $2k$-dimensional space), then the entire arrangement must be contained in a subspace whose dimension depends only on $k$ (and not on $n$). The theorem holds under the assumption that $V_a \cap V_b = \{0\}$ for every pair (otherwise it is false). This generalizes the Sylvester-Gallai theorem (or Kelly's theorem for complex numbers), which proves the …
We study representation stability in the sense of Church and Farb of sequences of cohomology groups of complements of arrangements of linear subspaces in real and complex space as $S_n$-modules. We consider arrangement of linear subspaces defined by sets of diagonal equalities $x_i = x_j$ and invariant under the action of $S_n$ permuting the coordinates. We provide bounds on the point when stabilization occurs and an alternative proof for the fact that stabilization happens. The latter is a special case of a very general stabilization result of Gadish and for the pure braid space the result is…
A subspace arrangement defined by intersections of hyperplanes of the braid arrangement can be encoded by an edge colored hypergraph. It turns out that the characteristic polynomial of this type of subspace arrangement is given by a generalized chromatic polynomial of the associated edge colored hypergraph. The main result of this paper supplies a sufficient condition for the existence of non-trivial Massey products of the subspace arrangements complex complement. This is accomplished by studying a spectral sequence associated to the Lie coalgebras of Sinha and Walter.
In 1978, Makai Jr. established a remarkable connection between the volume-product of a convex body, its maximal lattice packing density and the minimal density of a lattice arrangement of its polar body intersecting every affine hyperplane. Consequently, he formulated a conjecture that can be seen as a dual analog of Minkowski's fundamental theorem, and which is strongly linked to the well-known Mahler-conjecture. Based on the covering minima of Kannan & Lovász and a problem posed by Fejes Tóth, we arrange Makai Jr.'s conjecture into a wider context and investigate densities of lattice arrange…
Given a hyperplane arrangement in a complex vector space of dimension n, there is a natural associated arrangement of codimension k subspaces in a complex vector space of dimension k*n. Topological invariants of the complement of this subspace arrangement are related to those of the complement of the original hyperplane arrangement. In particular, if the hyperplane arrangement is fiber-type, then, apart from grading, the Lie algebra obtained from the descending central series for the fundamental group of the complement of the hyperplane arrangement is isomorphic to the Lie algebra of primitive…
Given a collection of $t$ subspaces in an $n$-dimensional $\mathbb{K} $-vector space $W$ we can associate to them $t$ vanishing ideals in the symmetric algebra $\mathcal{S}(W^*) = \mathbb{K}[x_1,x_2,\dots,x_n]$. As a subspace is defined by a set of linear equations, its vanishing ideal is generated by linear forms so it is a linear ideal. Conca and Herzog showed that the Castelnuovo-Mumford regularity of the product of $t$ linear ideals is equal to $t$. Derksen and Sidman showed that the Castelnuovo-Mumford regularity of the intersection of $t$ linear ideals is at most $t$ and they also showed…
We show that the cohomology algebra of the complement of a coordinate subspace arrangement in m-dimensional complex space is isomorphic to the cohomology algebra of Stanley-Reisner face ring of a certain simplicial complex on m vertices. (The face ring is regarded as a module over the polynomial ring on m generators.) Then we calculate the latter cohomology algebra by means of the standard Koszul resolution of polynomial ring. To prove these facts we construct an equivariant with respect to the torus action homotopy equivalence between the complement of a coordinate subspace arrangement and th…
For a natural number $c$, a $c$-arrangement is an arrangement of dimension $c$ subspaces satisfying the following condition: the sum of any subset of the subspaces has dimension a multiple of $c$. Matroids arising as normalized rank functions of $c$-arrangements are also known as multilinear matroids. We prove that it is algorithmically undecidable whether there exists a $c$ such that a given matroid has a $c$-arrangement representation, or equivalently whether the matroid is multilinear. It follows that certain network coding problems are also undecidable. In the proof, we introduce a general…
Let $\be$ be a Borel subalgebra of a complex simple Lie algebra $\g$. An ideal of $\be$ is called ad-nilpotent, if it is contained in $[\be,\be]$. The generators of an ad-nilpotent ideal give rise to an antichain in the poset of positive roots, and the whole theory can be expressed in a combinatorial fashion, in terms of antichains. The aim of this paper is to present a refinement of the enumerative theory of ad-nilpotent ideals for the case in which $\g$ has roots of different length. An antichain is called short, if it consists of short roots. We obtain, for short antichains, analogues of al…
Let A be a subspace arrangement and let chi(A,t) be the characteristic polynomial of its intersection lattice L(A). We show that if the subspaces in A are taken from L(B_n), where B_n is the type B Weyl arrangement, then chi(A,t) counts a certain set of lattice points. One can use this result to study the partial factorization of chi(A,t) over the integers and the coefficients of its expansion in various bases for the polynomial ring R[t]. Next we prove that the characteristic polynomial of any Weyl hyperplane arrangement can be expressed in terms of an Ehrhart quasi-polynomial for its affine …
To every subspace arrangement X we will associate symmetric functions P[X] and H[X]. These symmetric functions encode the Hilbert series and the minimal projective resolution of the product ideal associated to the subspace arrangement. They can be defined for discrete polymatroids as well. The invariant H[X] specializes to the Tutte polynomial T[X]. Billera, Jia and Reiner recently introduced a quasi-symmetric function F[X] (for matroids) which behaves valuatively with respect to matroid base polytope decompositions. We will define a quasi-symmetric function G[X] for polymatroids which has thi…
For any triple given by a positive integer n, a finite group G, and a faithful representation V of G, one can describe a subspace arrangement whose intersection lattice is a generalized Dowling lattice in the sense of Hanlon. In this paper we construct the minimal De Concini-Procesi wonderful model associated to this subspace arrangement and give a description of its boundary. Our aim is to point out the nice poset provided by the intersections of the irreducible components in the boundary, which provides a geometric realization of the nested set poset of this generalized Dowling lattice. It c…
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…
We study the intersection lattice of the arrangement $\mathcal{A}^G$ of subspaces fixed by subgroups of a finite linear group $G$. When $G$ is a reflection group, this arrangement is precisely the hyperplane reflection arrangement of $G$. We generalize the notion of finite reflection groups. We say that a group $G$ is generated (resp. strictly generated) in codimension $k$ if it is generated by its elements that fix point-wise a subspace of codimension at most $k$ (resp. precisely $k$). If $G$ is generated in codimension two, we show that the intersection lattice of $\mathcal{A}^G$ is atomic. …
A classic problem in matroid theory is to find subspace arrangements, specifically hyperplane and pseudosphere arrangements, whose intersection posets are isomorphic to a prescribed geometric lattice. Engström recently showed how to construct an infinite family of such subspace arrangements, indexed by the set of finite regular CW complexes. In this note, we compute the face numbers of these representations (in terms of the face numbers of the indexing complexes) and give upper bounds on the total number of faces in these objects. In particular, we show that, for a fixed rank, the total number…
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.
This paper presents a deterministic, strongly polynomial time algorithm for computing the matrix rank for a class of symbolic matrices (whose entries are polynomials over a field). This class was introduced, in a different language, by Lovász [Lov] in his study of flats in matroids, and proved a duality theorem putting this problem in $NP \cap coNP$. As such, our result is another demonstration where ``good characterization'' in the sense of Edmonds leads to an efficient algorithm. In a different paper Lovász [Lov79] proved that all such symbolic rank problems have efficient probabilistic algo…
A common generalization for the chromatic polynomial and the flow polynomial of a graph $G$ is the Tutte polynomial $T(G;x,y)$. The combinatorial meaning for the coefficients of $T$ was discovered by Tutte at the beginning of its definition. However, for a long time the combinatorial meaning for the values of $T$ is missing, except for a few values such as $T(G;i,j)$, where $1\leq i,j\leq 2$, until recently for $T(G;1,0)$ and $T(G;0,1)$. In this third one of a series of papers, we introduce product valuations, cartesian product arrangements, and multivariable characteristic polynomials, and ap…
We investigate the line arrangement that results from intersecting d complete flags in C^n. We give a combinatorial description of the matroid T_{n,d} that keeps track of the linear dependence relations among these lines. We prove that the bases of the matroid T_{n,3} characterize the triangles with holes which can be tiled with unit rhombi. More generally, we provide evidence for a conjectural connection between the matroid T_{n,d}, the triangulations of the product of simplices Delta_{n-1} x Δ_{d-1}, and the arrangements of d tropical hyperplanes in tropical (n-1)-space. Our work provides a …
We study the moduli space of $d$-dimensional linear subspaces contained in a fixed $(d+1)$-dimensional linear variety $X$, and its tropicalization. We prove that these moduli spaces are linear subspaces themselves, and thus their tropicalization is completely determined by their associated (valuated) matroids. We show that these matroids can be interpreted as the matroid of lines of the hyperplane arrangement corresponding to $X$, and generically are equal to a Dilworth truncation of the free matroid. In this way, we can describe combinatorially tropicalized Fano schemes and tropicalizations o…
The k-th Fitting ideal of the Alexander invariant B of an arrangement A of n complex hyperplanes defines a characteristic subvariety, V_k(A), of the complex algebraic n-torus. In the combinatorially determined case where B decomposes as a direct sum of local Alexander invariants, we obtain a complete description of V_k(A). For any arrangement A, we show that the tangent cone at the identity of this variety coincides with R^1_k(A), one of the cohomology support loci of the Orlik-Solomon algebra. Using work of Arapura and Libgober, we conclude that all positive-dimensional components of V_k(A) a…
We construct bundles $E_k(\A,\F) \to M$ over the complement $M$ of a complex hyperplane arrangement \A, depending on an integer $k \geq 1$ and a set $\F=\{f_1, \ldots, f_μ\}$ of continuous functions $f_i \colon M \to \C$ whose differences are nonzero on $M$, generalizing the configuration space bundles arising in the Lawrence-Krammer-Bigelow representation of the pure braid group. We display such families \F\ for rank two arrangements, reflection arrangements of types $A_\ell$, $B_\ell$, $D_\ell$, $F_4$, and for arrangements supporting multinet structures with three classes, with the resulting…
We consider hyperplane arrangements generated by generic points and study their intersection lattices. These arrangements are known to be equivalent to discriminantal arrangements. We show a fundamental structure of the intersection lattices by decomposing the poset ideals as direct products of smaller lattices corresponding to smaller dimensions. Based on this decomposition we compute the Möbius functions of the lattices and the characteristic polynomials of the arrangements up to dimension six.
Using the topological technique of diagrams of spaces, we calculate the homology of the union and the complement of finite arrangements of subspaces of the form $D + SP^{n-d}(X)$ in symmetric products $SP^n(X)$ where $D\in SP^d(X)$. As an application we include a computation of the homology of the homotopy end space of the open manifold $SP^n(M_{g,k})$, where $M_{g,k}$ is a Riemann surface of genus $g$ punctured at $k$ points, a problem which was originally motivated by the study of commutative $(m+k,m)$-groups.
A finite simple graph $Γ$ determines a quotient $P_Γ$ of the pure braid group, called a graphic arrangement group. We analyze homomorphisms of these groups defined by deletion of sets of vertices, using methods developed in prior joint work with R. Randell. We show that, for a $K_4$-free graph $Γ$, a product of deletion maps is injective, embedding $P_Γ$ in a product of free groups. Then $P_Γ$ is residually free, torsion-free, residually torsion-free nilpotent, and acts properly on a CAT(0) cube complex. We also show $P_Γ$ is of homological finiteness type $F_{m-1}$, but not $F_m$, where $m$ i…
Let \A be an affine hyperplane arrangement in $\C^\ell$ with complement $U$. Let $f_1, \..., f_n$ be linear polynomials defining the hyperplanes of \A, and $A^\cdot$ the algebra of differential forms generated by the 1-forms $d \log f_1, \..., d \log f_n$. To each $l \in \C^n$ we associate the master function $Φ=Φ_l = \prod_{i=1}^n f_i^{l_i}$ on $U$ and the closed logarithmic 1-form $ω= d \log Φ$. We assume $ω$ is an element of a rational linear subspace $D$ of $A^1$ of dimension $q>1$ such that the multiplication map $\bigwedge^k(D) \to A^k$ is zero for $p<k\leq q$. With this assumption, we p…
