On topological and geometric (19₄)configurations

An (n_k)configuration is a set of n points and n lines such that each point lies on k lines while each line contains k points. The configuration is geometric, topological, or combinatorial depending on whether lines are considered to be straight lines, pseudolines, or just combinatorial lines. The existence and enumeration of (n_k)configurations for a given k has been subject to active research. A current front of research concerns geometric (n₄)configurations: it is now known that geometric (n₄)configurations exist for all n≥18, apart from sporadic exceptional cases. In this paper, we settle by computational techniques the first open case of (19₄)configurations: we obtain all topological (19₄)configurations among which none are geometrically realizable.