It is known since the works of Zariski in the early 40s that desingularization of varieties along valuations (called local uniformization of valuations) can be considered as the local part of the desingularization problem. It is still an open problem if local uniformization exists in positive characteristic and dimension larger than three. In this paper, we prove that Zariski local uniformization of algebraic varieties is always possible after a purely inseparable extension of the field of rational functions, and therefore any valuation can be uniformized by a purely inseparable alteration.
Bibliographical noteFunding Information:
1 Parts of this work were done when I was visiting Max Planck Institute for Mathematics at Bonn and the Institute for Advanced Study at Princeton, and I am thankful to them for the hospitality. In IAS I was supported by NSF grant DMS-0635607. In the end of this project my research was supported by Marie Curie International Reintegration Grant 268182 within the 7th European Community Framework Programme.
- Inseparable local uniformization