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- 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 , 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. 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.) 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 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. simplified Abhyankar's long proof. Cossart and Piltant extended Abhyankar's proof of local uniformization of 3-folds to the remaining characteristics 2, 3, and 5. 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 2019) an open problem. (en)
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- Cossart (en)
- Piltant (en)
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- 2008 (xsd:integer)
- 2009 (xsd:integer)
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- 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 , 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. (en)
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- Local uniformization (en)
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