Nash modification on toric surfaces

TL;DR

This paper investigates the Nash modification process on toric surfaces, showing it terminates under certain conditions and providing bounds on the number of steps needed, with implications for local uniformization.

Contribution

It demonstrates the termination of Nash modification on toric surfaces for specific affine charts and establishes bounds on the iteration steps required.

Findings

Nash modification algorithm stops for certain affine charts on toric surfaces.

Provides bounds on the number of steps for the Nash modification to terminate.

Ensures local uniformization along specific valuations after finite steps.

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 $\C(x_1,x_2)$ 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 $\nu$.