Toric geometry and the Semple–Nash modification

This paper proposes some material towards a theory of general toric varieties without the assumption of normality. Their combinatorial description involves a fan to which is attached a set of semigroups subjected to gluing-up conditions. In particular it contains a combinatorial construction of the blowing up of a sheaf of monomial ideals on a toric variety. In the second part this is used to show that iterating the Semple–Nash modification or its characteristic-free avatar provides a local uniformization of any monomial valuation of maximal rank dominating a point of a toric variety.