The Toroidal Embedding Arising from an Irrational Fan

A toroidal embedding is defined which does not assume the fan consists of rational cones. For a rational fan, the toroidal embedding is the usual toric variety. If the fan is not rational, the toroidal embedding is in general a quasicompact noetherian locally ringed space which is not a scheme. A divisor theory exists and a class group is defined. A second construction is also carried out which mimics the gluing construction of the usual toric variety, but which makes no reference to a lattice. The resulting scheme is separated but infinite dimensional. The Picard group is described in terms of the group of real valued locally linear support functions on the fan and the Brauer group is shown to be trivial. Many examples are given throughout the paper; in particular, it is shown that there is associated to a real hyperplane arrangement of full rank a toroidal embedding.