In their solution to the orchard-planting problem, Green and Tao established a structure theorem which proves that in a line arrangement in the real projective plane with few double points, most lines are tangent to the dual curve of a cubic curve. We provide geometric arguments to prove that in the case of a simplicial arrangement, the aforementioned cubic curve cannot be irreducible. It follows that Grünbaum's conjectural asymptotic classification of simplicial arrangements holds under the additional hypothesis of a linear bound on the number of double points.
Simplicial arrangements are classical objects in discrete geometry. Their classification remains an open problem but there is a list conjectured to be complete at least for rank three. A further important class in the theory of hyperplane arrangements with particularly nice geometric, algebraic, topological, and combinatorial properties are the supersolvable arrangements. In this paper we give a complete classification of supersolvable simplicial arrangements (in all ranks). For each fixed rank, our classification already includes almost all known simplicial arrangements. Surprisingly, for irr…
We compute all isomorphism classes of simplicial arrangements in the real projective plane with up to 27 lines. It turns out that Grünbaums catalogue is complete up to 27 lines except for four new arrangements with 22, 23, 24, 25 lines, respectively. As a byproduct we classify simplicial arrangements of pseudolines with up to 27 lines. In particular, we disprove Grünbaums conjecture about unstretchable arrangements with at most 16 lines, and prove the conjecture that any simplicial arrangement with at most 14 pseudolines is stretchable.
We introduce the notion of a Tits arrangement on a convex open cone as a special case of (infinite) simplicial arrangements. Such an object carries a simplicial structure similar to the geometric representation of Coxeter groups. The standard constructions of subarrangements and restrictions, which are known in the case of finite hyperplane arrangements, work as well in this more general setting.
We introduce the simple notion of a "crystallographic arrangement" and prove a one-to-one correspondence between these arrangements and the connected simply connected Cartan schemes for which the real roots are a finite root system (up to equivalence on both sides). We thus obtain a more accessible definition for this very large subclass of the class of simplicial arrangements for which a complete classification is known.
In this paper, we study simplicial hyperplane arrangements in real projective $3$-space. We give a necessary condition for the characteristic polynomial to have only real roots, valid also for non-simplicial arrangements. As application, we obtain combinatorial inequalities which are satisfied for arrangements with splitting polynomial. This allows us to prove that there are only finitely many different isomorphism classes of simply laced simplicial arrangements whose characteristic polynomials split over $\mathbb{R}$. We also provide an updated version of a catalogue published by Grünbaum and…
An arrangement of hyperplanes is strongly inscribable if it has an inscribed (or ideal hyperbolic) zonotope. We characterize inscribed zonotopes and prove that the family of strongly inscribable arrangements is closed under restriction and localization. Moreover, we show that (strongly) inscribable arrangements are simplicial. We conjecture that only reflection arrangements and their restrictions are strongly inscribable and we verify our conjecture in rank-$3$ using the conjecturally complete list of irreducible simplicial rank-$3$ arrangements.
We introduce a combinatorial characterization of simpliciality for arrangements of hyperplanes. We then give a sharp upper bound for the number of hyperplanes of such an arrangement in the projective plane over a finite field, and present some series of arrangements related to the known arrangements in characteristic zero. We further enumerate simplicial arrangements with given symmetry groups. Finally, we determine all finite complex reflection groups affording combinatorially simplicial arrangements. It turns out that combinatorial simpliciality coincides with inductive freeness for finite c…
A catalogue of simplicial hyperplane arrangements was first given by Grünbaum in 1971. These arrangements naturally generalize finite Coxeter arrangements and the weak order through the poset of regions. For simplicial arrangements, posets of regions are in fact lattices. We update Grünbaum's catalogue, providing normals and invariants for all known sporadic simplicial arrangements with up to 37 lines. The weak order is known to be congruence normal, and congruence normality for simplicial arrangements can be determined using polyhedral cones called shards. In this article, we provide addition…
We study the combinatorics of pseudoline arrangements in the real projective plane. Our focus lies on two classes of arrangements: simplicial arrangements and arrangements whose characteristic polynomials have only real roots. We derive inequalities involving the $t$-vectors of the arrangements in consideration. As application, we obtain some finiteness and classification results. Moreover, we are able to prove the Dirac Motzkin Conjecture for real pseudoline arrangements whose charateristic polynomials split over $\mathbb{R}$.
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.
We introduce a greedy algorithm optimizing arrangements of lines with respect to a property. We apply this algorithm to the case of simpliciality: it recovers all known simplicial arrangements of lines in a very short time and also produces a yet unknown simplicial arrangement with 35 lines. We compute a (certainly incomplete) database of combinatorially simplicial complex arrangements of hyperplanes with up to 50 lines. Surprisingly, it contains several examples whose matroids have an infinite space of realizations up to projectivities.
We extend the notion of face rings of simplicial complexes and simplicial posets to the case of finite-length (possibly infinite) simplicial posets with a group action. The action on the complex induces an action on the face ring, and we prove that the ring of invariants is isomorphic to the face ring of the quotient simplicial poset under a mild condition on the group action. We also identify a class of actions on simplicial complexes that preserve the homotopical Cohen-Macaulay property under quotients. When the acted-upon poset is the independence complex of a semimatroid, the $h$-polynomia…
Two arrangements with the same combinatorial intersection lattice but whose complements have different fundamental groups are called a Zariski pair. This work finds that there are at most nine such pairs amongst all ten line arrangements whose intersection points are doubles or triples. This result is obtained by considering the moduli space of a given configuration table which describes the intersection lattice. A complete combinatorial classification is given of all arrangements of this type under a suitable assumption, producing a list of seventy-one described in a table, most of which do n…
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 supersolva…
We prove that the mod $2$ Betti numbers of double coverings of a complex hyperplane arrangement complement are combinatorially determined. The proof is based on a relation between the mod $2$ Aomoto complex and the transfer long exact sequence. Applying the above result to the icosidodecahedral arrangement ($16$ planes in the three dimensional space related to the icosidodecahedron), we conclude that the first homology of the Milnor fiber has non-trivial $2$-torsion.
Using the classification of finite Weyl groupoids we prove that crystallographic arrangements, a large subclass of the class of simplicial arrangements which was recently defined, are hereditarily inductively free. In particular, all crystallographic reflection arrangements are hereditarily inductively free, among them the arrangement of type $E_8$. With little extra work we prove that also all Coxeter arrangements are inductively free.
We describe an incremental algorithm to enumerate the isomorphism classes of double pseudoline arrangements. The correction of our algorithm is based on the connectedness under mutations of the spaces of one-extensions of double pseudoline arrangements, proved in this paper. Counting results derived from an implementation of our algorithm are also reported.
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 …
The main purpose of the note is to show that Yoshinaga's arrangement of $18$ lines having $48$ triple and $9$ double intersection points leads to a new (short) series of non-containment examples for $I^{(3)} \subset I^{2}$, the question studied by Harbourne and Huneke.
We study a non-trivial extreme case of the orchard problem for $12$ pseudolines and we provide a complete classification of pseudoline arrangements having $19$ triple points and $9$ double points. We have also classified those that can be realized with straight lines. They include new examples different from the known example of Böröczky. Since Melchior's inequality also holds for arrangements of pseudolines, we are able to deduce that some combinatorial point-line configurations cannot be realized using pseudolines. In particular, this gives a negative answer to one of Grünbaum's problems. We…
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 …
We introduce a new model for random simplicial complexes which with high probability generates a complex that has a simply-connected double cover. Hence we develop a model for random simplicial complexes with fundamental group $\mathbb{Z}/2\mathbb{Z}$. We establish results about the typical asymptotic topology of these complexes. As a consequence we give bounds for the dimension $d$ such that $\mathbb{Z}/2\mathbb{Z}$-equivariant maps from the double cover to $\mathbb{R}^d$ have zeros with high probability, thus establishing a random Borsuk--Ulam theorem. We apply this to derive a structural re…
