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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 Zieglers 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 $dgeq 9$ of lines, and any number $n_3geq 6$ 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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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 Teraos 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 Teraos 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.
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.
We reformulate a fundamental result due to Cook, Harbourne, Migliore and Nagel on the existence and irreduciblity of unexpected plane curves of a set of points $Z$ in $mathbb{P}^2$, using the minimal degree of a Jacobian syzygy of the defining equation for the dual line arrangement $mathcal A_Z$. Several applications of this new approach are given. In particular, we show that the irreducible unexpected quintics may occur only when the set $Z$ has the cardinality equal to 11 or 12, and describe five cases where this happens.
We introduce the package textbf{arrangements} for the software CoCoA. This package provides a data structure and the necessary methods for working with hyperplane arrangements. In particular, the package implements methods to enumerate many commonly studied classes of arrangements, perform operations on them, and calculate various invariants associated to them.
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. Tohu 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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