  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 $\C(x_1,x_2)$ be the field of rational functions of a toric surface. Then
our result implies that if $\nu:\C(x_1,x_2)\rightarrow\Gamma$ is any valuation
centered on the toric surface and such that $\nu(x_1)\neq\lambda\nu(x_2)$ for
all $\lambda\in\R\setminus\Q$, then a finite iteration of Nash modification
gives local uniformization along $\nu$.

Algebraic Geometry