A generalized logarithmic module and duality of Coxeter multiarrangements

We introduce a new definition of a generalized logarithmic module of multiarrangements by uniting those of the logarithmic derivation and the differential modules. This module is realized as a logarithmic derivation module of an arrangement of hyperplanes with a multiplicity consisting of both positive and negative integers. We consider several properties of this module including Saito's criterion and reflexivity. As applications, we prove a shift isomorphism and duality of some Coxeter multiarrangements by using the primitive derivation.Comment: 17 page

Roots of characteristic polynomials and intersection points of line arrangements

We study a relation between roots of characteristic polynomials and intersection points of line arrangements. Using these results, we obtain a lot of applications for line arrangements. Namely, we give (i) a generalized addition theorem for line arrangements, (ii) a generalization of Faenzi-Vall\`{e}s' theorem over a field of arbitrary characteristic, (iii) a partial result on the conjecture of Terao of line arrangements, and (iv) a new sufficient condition for freeness over finite fields. Also, a higher dimensional version of main results is considered.Comment: 18 pages (ver.1). 20 pages (ver. 2). Example 1.4 (1) and Theorem 1.6 are added. 18 pages (ver. 3). Section 7 is adde

The freeness of Shi-Catalan arrangements

Let $W$ be a finite Weyl group and \A be the corresponding Weyl arrangement. A deformation of \A is an affine arrangement which is obtained by adding to each hyperplane H\in\A several parallel translations of $H$ by the positive root (and its integer multiples) perpendicular to $H$. We say that a deformation is $W$-equivariant if the number of parallel hyperplanes of each hyperplane H\in \A depends only on the $W$-orbit of $H$. We prove that the conings of the $W$-equivariant deformations are free arrangements under a Shi-Catalan condition and give a formula for the number of chambers. This generalizes Yoshinaga's theorem conjectured by Edelman-Reiner.Comment: 12 page