Euclidean -systems and real PK arrangements
arXiv:2607.04859
Abstract
We establish a correspondence between two structures arising in the geometry of hyperplane arrangements: Euclidean -systems and real polyhedral Kähler (PK) arrangements. We prove that every irreducible Euclidean -system determines a real PK arrangement, and conversely that every real PK arrangement arises this way. As a consequence, we show that, up to equivalence, there are exactly three irreducible rank-three Euclidean -systems whose vectors have equal length; their arrangements are the mirrors of the reflection groups of the regular tetrahedron, cube, and icosahedron. The correspondence also yields a description of the moduli space of Euclidean -systems in a fixed projective class: it is homeomorphic to the relative interior of a polytope. We also give a direct proof that the hyperplane arrangement associated with a Euclidean -system is simplicial. Among the currently known simplicial line arrangements, we identify precisely those that arise from -systems. As a consequence, we prove that the Schreiber--Veselov catalog is complete for irreducible rank-three Euclidean -systems with at most vectors.
30 pages. Added classification of rank- irreducible -systems whose vectors have equal length