Local uniformization of valuations

Establish local uniformization for arbitrary valuations of function fields, thereby showing that every valuation is centered on a regular point of a variety with the given function field.

Background

The paper studies Frobenius splitting of valuation rings and proves that, over an F-finite ground field, Frobenius-split valuation rings correspond precisely to Abhyankar valuations. Abhyankar valuations are known to admit local uniformization: they can be centered on regular points of suitable varieties. However, the authors distinguish this special case from the broader local-uniformization problem for arbitrary valuations of function fields.

Resolving local uniformization in full generality would extend the known regular-center results from Abhyankar valuations to all valuations and would address a major unresolved issue in valuation theory and birational geometry. The paper does not attempt to solve this problem; it only notes the established positive result for Abhyankar valuations.

References

Moreover, while the outstanding problem of local uniformization of valuations remains open, Abhyankar valuations on function fields always admit a local uniformization, i.e., they are centered on a {\it regular} point of a variety $X/k$ with function field $K$ .

— Frobenius splitting of valuation rings revisited  (2610.11032 - Datta et al., 8 Oct 2026) in Section 1, Introduction