Local uniformization

In algebraic geometry, local uniformization is a weak form of resolution of singularities, stating roughly that a variety can be desingularized near any valuation. 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. 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.

Local uniformization in positive characteristic for varieties of dimension at least 4 is (as of 2013) an open problem.

References

• Abhyankar, Shreeram (1956), "Local uniformization on algebraic surfaces over ground fields of characteristic p≠0", Annals of Mathematics. Second Series, 63 (3): 491–526, doi:10.2307/1970014, JSTOR 1970014, MR 0078017
• Abhyankar, Shreeram S. (1966), Resolution of singularities of embedded algebraic surfaces, Acad. Press, doi:10.1007/978-3-662-03580-1, ISBN 3-540-63719-2 (1998 2nd edition)
• Cossart, Vincent; Piltant, Olivier (2008), "Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings", Journal of Algebra, 320 (3): 1051–1082, doi:10.1016/j.jalgebra.2008.03.032, MR 2427629
• Cossart, Vincent; Piltant, Olivier (2009), "Resolution of singularities of threefolds in positive characteristic. II", Journal of Algebra, 321 (7): 1836–1976, doi:10.1016/j.jalgebra.2008.11.030, MR 2494751
• Cutkosky, Steven Dale (2004), Resolution of Singularities, Providence, RI: American Math. Soc., ISBN 0-8218-3555-6
• Cutkosky, Steven Dale (2009), "Resolution of singularities for 3-folds in positive characteristic", Amer. J. Math., 131 (1): 59–127, JSTOR 40068184, MR 2488485
• Zariski, Oscar (1939), "The reduction of the singularities of an algebraic surface", Ann. Of Math. (2), 40 (3): 639–689, doi:10.2307/1968949, JSTOR 1968949
• Zariski, Oscar (1940), "Local uniformization on algebraic varieties", Ann. of Math. (2), 41: 852–896, JSTOR 1968864, MR 0002864
• Zariski, Oscar (1944), "The compactness of the Riemann manifold of an abstract field of algebraic functions", Bulletin of the American Mathematical Society, 50: 683–691, doi:10.1090/S0002-9904-1944-08206-2, ISSN 0002-9904, MR0011573
• Zariski, Oscar (1944), "Reduction of the singularities of algebraic three dimensional varieties", Ann. Of Math. (2), 45 (3): 472–542, doi:10.2307/1969189, JSTOR 1969189, MR 0011006