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This paper proposes some material towards a theory of general toric varietieswithout the assumption of normality. Their combinatorial description involves afan 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 asheaf of monomial ideals on a toric variety. In the second part it is shownthat over an algebraically closed base field of zero characteristic theSemple-Nash modification of a general toric variety is isomorphic to theblowing up of the sheaf of logarithmic jacobian ideals and that in anycharacteristic this blowing-up is an isomorphism if and only if the toricvariety is non singular. In the second part we prove that orders on the latticeof monomials (toric valuations) of maximal rank are uniformized by iteratedSempla-Nash modifications.

Toric Geometry and the Semple-Nash modification