# Toric Geometry and the Semple-Nash modification

## TL;DR

This paper develops a combinatorial framework for general toric varieties, extending the Semple-Nash modification theory to non-normal cases and linking it to blowing up monomial ideals and valuations.

## Contribution

It introduces a combinatorial description of toric varieties without normality and relates the Semple-Nash modification to blowing up logarithmic jacobian ideals.

## Key findings

- Semple-Nash modification is isomorphic to blowing up logarithmic jacobian ideals in characteristic zero.
- Blowing-up of monomial ideal sheaves generalizes to non-normal toric varieties.
- Iterated Semple-Nash modifications uniformize toric valuations of maximal rank.

## Abstract

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 it is shown that over an algebraically closed base field of zero characteristic the Semple-Nash modification of a general toric variety is isomorphic to the blowing up of the sheaf of logarithmic jacobian ideals and that in any characteristic this blowing-up is an isomorphism if and only if the toric variety is non singular. In the second part we prove that orders on the lattice of monomials (toric valuations) of maximal rank are uniformized by iterated Sempla-Nash modifications.