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- Local_uniformization abstract "In algebraic geometry, local uniformization is a weak form of resolution of singularities, stating roughly that a variety can be desingularized near any valuation, or in other words that the Zariski–Riemann space of the variety is in some sense nonsingular. Local uniformization was introduced by Zariski (1939, 1940), who separated out the problem of resolving the singularities of a variety into the problem of local uniformization and the problem of combining the local uniformizations into a global desingularization.Local uniformization of a variety at a valuation of its function field means finding a projective model of the variety such that the center of the valuation is non-singular. This is weaker than resolution of singularities: if there is a resolution of singularities then this is a model such that the center of every valuation is non-singular. Zariski (1944) proved that if one can show local uniformization of a variety then one can find a finite number of models such that every valuation has a non-singular center on at least one of these models. To complete a proof of resolution of singularities it is then sufficient to show that one can combine these finite models into a single model, but this seems rather hard. (Local uniformization at a valuation does not directly imply resolution at the center of the valuation: roughly speaking; it only implies resolution in a sort of "wedge" near this point, and it seems hard to combine the resolutions of different wedges into a resolution at a point.)Zariski (1940) proved local uniformization of varieties in any dimension over fields of characteristic 0, and used this to prove resolution of singularities for varieties in characteristic 0 of dimension at most 3. Local uniformization in positive characteristic seems to be much harder. Abhyankar (1956, 1966) proved local uniformization in all characteristic for surfaces and in characteristics at least 7 for 3-folds, and was able to deduce global resolution of singularities in these cases from this. Cutkosky (2009) simplified Abhyankar's long proof. Cossart and Piltant (2008, 2009) extended Abhyankar's proof of local uniformization of 3-folds to the remaining characteristics 2, 3, and 5. Temkin (2013) showed that it is possible to find a local uniformization of any valuation after taking a purely inseparable extension of the function field. Local uniformization in positive characteristic for varieties of dimension at least 4 is (as of 2013) an open problem.".
- Local_uniformization wikiPageID "40061096".
- Local_uniformization wikiPageLength "5485".
- Local_uniformization wikiPageOutDegree "9".
- Local_uniformization wikiPageRevisionID "626965791".
- Local_uniformization wikiPageWikiLink Annals_of_Mathematics.
- Local_uniformization wikiPageWikiLink Bulletin_of_the_American_Mathematical_Society.
- Local_uniformization wikiPageWikiLink Category:Algebraic_geometry.
- Local_uniformization wikiPageWikiLink Category:Singularity_theory.
- Local_uniformization wikiPageWikiLink Center_(valuation_ring).
- Local_uniformization wikiPageWikiLink Journal_of_Algebra.
- Local_uniformization wikiPageWikiLink Resolution_of_singularities.
- Local_uniformization wikiPageWikiLink Valuation_ring.
- Local_uniformization wikiPageWikiLink Zariski–Riemann_space.
- Local_uniformization wikiPageWikiLinkText "Local uniformization".
- Local_uniformization wikiPageWikiLinkText "local uniformization".
- Local_uniformization hasPhotoCollection Local_uniformization.
- Local_uniformization last "Cossart".
- Local_uniformization last "Piltant".
- Local_uniformization wikiPageUsesTemplate Template:Citation.
- Local_uniformization wikiPageUsesTemplate Template:Eom.
- Local_uniformization wikiPageUsesTemplate Template:Harvs.
- Local_uniformization wikiPageUsesTemplate Template:Harvtxt.
- Local_uniformization year "2008".
- Local_uniformization year "2009".
- Local_uniformization subject Category:Algebraic_geometry.
- Local_uniformization subject Category:Singularity_theory.
- Local_uniformization hypernym Form.
- Local_uniformization comment "In algebraic geometry, local uniformization is a weak form of resolution of singularities, stating roughly that a variety can be desingularized near any valuation, or in other words that the Zariski–Riemann space of the variety is in some sense nonsingular.".
- Local_uniformization label "Local uniformization".
- Local_uniformization sameAs m.0wdt5cf.
- Local_uniformization sameAs Q17098591.
- Local_uniformization sameAs Q17098591.
- Local_uniformization wasDerivedFrom Local_uniformization?oldid=626965791.
- Local_uniformization isPrimaryTopicOf Local_uniformization.