Local uniformization

Template:Short description In algebraic geometry, local uniformization is a weak form of resolution of singularities, stating that a variety can be desingularized near any valuation, or in other words that the Zariski–Riemann space of the array is in some sense non-singular. Local uniformization was introduced by Zariski (1939, 1940), who separated 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. It 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 (1956, 1966) proved local uniformization in all characteristics 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 (2008, 2009) 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

• 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, Springer Monographs in Mathematics, 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" (PDF), Journal of Algebra, 321 (7): 1836–1976, doi:10.1016/j.jalgebra.2008.11.030, MR 2494751
• Cutkosky, Steven Dale (2009), "Resolution of singularities for 3-folds in positive characteristic", Amer. J. Math., 131 (1): 59–127, arXiv:math/0606530, doi:10.1353/ajm.0.0036, JSTOR 40068184, MR 2488485, S2CID 2139305
• Temkin, Michael (2013), "Inseparable local uniformization", J. Algebra, 373: 65–119, arXiv:0804.1554, doi:10.1016/j.jalgebra.2012.09.023, MR 2995017, S2CID 115167009
• 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 (4): 852–896, doi:10.2307/1968864, JSTOR 1968864, MR 0002864
• Zariski, Oscar (1944a), "The compactness of the Riemann manifold of an abstract field of algebraic functions", Bulletin of the American Mathematical Society, 50 (10): 683–691, doi:10.1090/S0002-9904-1944-08206-2, ISSN 0002-9904, MR 0011573
• Zariski, Oscar (1944b), "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