On the local uniformization problem

TL;DR

This paper introduces the local uniformization problem, discusses reduction to rank one valuations, and extends the reduction to rings with zero divisors, advancing understanding in algebraic geometry.

Contribution

It proves that local uniformization for certain valuations can be reduced to the rank one case, including non-reduced rings, extending previous results.

Findings

Reduction of local uniformization to rank one valuations.

Extension of reduction to rings with zero divisors.

Framework applicable to non-reduced rings.

Abstract

In this paper we give a short introduction to the local uniformization problem. This follows a similar line as the one presented by the second author in his talk at ALANT 3. We also discuss our paper on the reduction of local uniformization to the rank one case. In that paper, we prove that in order to obtain local uniformization for valuations centered at objects of a subcategory of the category of noetherian integral domains, it is enough to prove it for rank one valuations centered at objects of the same category. We also announce an extension of this work which was partially developed during ALANT 3. This extension says that the reduction mentioned above also works for noetherian rings with zero divisors (including the case of non-reduced rings).