Inseparable local uniformization

Michael Temkin

doi:10.1016/j.jalgebra.2012.09.023

Inseparable local uniformization

Michael Temkin^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

31 Scopus citations

Abstract

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.

Original language	English
Pages (from-to)	65-119
Number of pages	55
Journal	Journal of Algebra
Volume	373
DOIs	https://doi.org/10.1016/j.jalgebra.2012.09.023
State	Published - 1 Jan 2013

Bibliographical note

Funding 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.

Keywords

Desingularization
Inseparable local uniformization

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10.1016/j.jalgebra.2012.09.023

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abstract = "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.",

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AB - 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.

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Inseparable local uniformization

Abstract

Bibliographical note

Keywords

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