Nash modification on toric surfaces

Abstract

It has been recently shown that the iteration of Nash modification on not necessarily normal toric varieties corresponds to a purely combinatorial algorithm on the generators of the semigroup associated to the toric variety. We will show that for toric surfaces this algorithm stops for certain choices of affine charts of the Nash modification. In addition, we give a bound on the number of steps required for the algorithm to stop in the cases we consider. Let $\mathbb{C }(x_1,x_2)$ be the field of rational functions of a toric surface. Then our result implies that if $\nu :\mathbb{C }(x_1,x_2)\rightarrow \Gamma$ is any valuation centered on the toric surface and such that $\nu (x_1)\ne \lambda \nu (x_2)$ for all $\lambda \in \mathbb{R }\setminus \mathbb{Q }$, then a finite iteration of Nash modification gives local uniformization along $\nu$.