Local Uniformization
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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 In algebraic geometry, a Zariski–Riemann space or Zariski space of a subring ''k'' of a field ''K'' is a locally ringed space whose points are valuation rings containing ''k'' and contained in ''K''. They generalize the Riemann surface of a c ...
of the variety is in some sense nonsingular. Local uniformization was introduced by , 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 Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...
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. 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. 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.


References

* * (1998 2nd edition) * * * * * * * *


External links

*{{SpringerEOM, title=Local uniformization Algebraic geometry Singularity theory