Inseparable local uniformization

By M. Temkin. 2008

69 citations recorded by the source corpus.

It is known since the works of Zariski in early 40ies 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, i.e. any valuation can be uniformized by a purely inseparable alteration.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Rehabilitation

View Paper: https://www.semanticscholar.org/paper/a84523219a5c2dcc18a612002638ebc92418f7aa

How researchers see this paper

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In Computational Theory and Mathematics

1. L. Bary‐Soroker
2. Arno Fehm
3. Kiran S. Kedlaya
4. Alan G. B. Lauder
5. T. Mochizuki
6. Ruochuan Liu
7. David Lubicz
8. Reynald Lercier

In Geometry and Topology

1. L. Bary‐Soroker
2. Arno Fehm
3. Kiran S. Kedlaya
4. Ruochuan Liu
5. Peter Scholze
6. Alan G. B. Lauder
7. T. Mochizuki
8. Jonathan Pottharst

In Rehabilitation

1. L. Bary‐Soroker
2. Arno Fehm
3. Kiran S. Kedlaya

Citations in the corpus

• On fields of totally $\mathfrak{S}$-adic numbers. With an appendix by Florian Pop (2012, background): “Temkin (see [80]) on inseparable local uniformization, which in particular proves local uniformization for Abhyankar valuations of algebraic function fields and also uses toroidal methods.”
• Good formal structures for flat meromorphic connections, I: Surfaces (2008, methodology): “We will instead use the valuation-theoretic approach that we introduced in our recent work on the problem of semistable reduction for overconvergent F -isocrystals [5, 6, 7, 8]; a similar approach appears in Temkin’s proof of inseparable local uniformization [17].”
• Good formal structures for flat meromorphic connections, II: Excellent schemes (2010, methodology): “We overcome this difficulty using a new idea, drawn from our work on semistable reduction for overconvergent F -isocrystals [22, 23, 24, 26], and from Temkin’s proof of inseparable local uniformization for function fields in positive characteristic [41].”
• Resolution of singularities of arithmetical threefolds (2014, unknown): “Temkin’s purely inseparable Local Uniformization Theorem [80] Theorem 1.”