Simplicial arrangements with special vertex

Shi-Catalan and affine Weyl arrangements are obtained in a similar way: they consist of a Weyl arrangement embedded into a higher dimensional space together with shifted copies. We consider the more general case when the Weyl arrangement is replaced by an arbitrary finite arrangement of hyperplanes. Such an arrangement is called an arrangement with special vertex. As our main result we prove that the number of parallel classes of lines in certain simplicial arrangements with special vertex is at most 12.