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In `F33f`_`[algebraic geometry`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_geometry]`_`f, `!local uniformization`! is a weak form of `F33f`_`[resolution of singularities`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Resolution_of_singularities]`_`f, stating that a variety can be desingularized near any `F33f`_`[valuation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Valuation_(algebra)]`_`f, or in other words that the `F33f`_`[Zariski–Riemann space`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Zariski–Riemann_space]`_`f of the array is in some sense non-singular. Local uniformization was introduced by Zariski (`F33f`_`[1939`#citerefzariski1939]`_`f, `F33f`_`[1940`#citerefzariski1940]`_`f), 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 `F33f`_`[variety`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_variety]`_`f at a valuation of its `F33f`_`[function field`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_field_of_an_algebraic_variety]`_`f means finding a projective model of the variety such that the `F33f`_`[center`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Center_(valuation_ring)]`_`f 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. `F33f`_`[Zariski (1944b)`#citerefzariski1944b]`_`f 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.)

`F33f`_`[Zariski (1940)`#citerefzariski1940]`_`f proved local uniformization of varieties in any dimension over fields of `F33f`_`[characteristic`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Characteristic_(algebra)]`_`f 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 (`F33f`_`[1956`#citerefabhyankar1956]`_`f, `F33f`_`[1966`#citerefabhyankar1966]`_`f) 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. `F33f`_`[Cutkosky (2009)`#citerefcutkosky2009]`_`f simplified Abhyankar's long proof. Cossart and Piltant (`F33f`_`[2008`#citerefcossartpiltant2008]`_`f, `F33f`_`[2009`#citerefcossartpiltant2009]`_`f) extended Abhyankar's proof of local uniformization of 3-folds to the remaining characteristics 2, 3, and 5. `F33f`_`[Temkin (2013)`#citereftemkin2013]`_`f showed that it is possible to find a local uniformization of any valuation after taking a `F33f`_`[purely inseparable extension`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Purely_inseparable_extension]`_`f of the function field.

Local uniformization in positive characteristic for varieties of dimension at least 4 is (as of 2019) an open problem.

>>Contents

• `F0af`_`[References`#references]`_`f

• `F0af`_`[External links`#external-links]`_`f

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>>References

• `:citerefabhyankar1956`aAbhyankar, Shreeram (1956), "Local uniformization on algebraic surfaces over ground fields of characteristic `*p`*≠0", `*`F33f`_`[Annals of Mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Annals_of_Mathematics]`_`f`*, Second Series, `!63`! (3): 491–526, `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.2307/1970014, `F33f`_`[JSTOR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=JSTOR_(identifier)]`_`f 1970014, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 0078017

• `:citerefabhyankar1966`a`F33f`_`[Abhyankar, Shreeram S.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S._S._Abhyankar]`_`f (1966), `*Resolution of singularities of embedded algebraic surfaces`*, Springer Monographs in Mathematics, Acad. Press, `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1007/978-3-662-03580-1, `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 3-540-63719-2 (1998 2nd edition)

• `:citerefcossartpiltant2008`aCossart, Vincent; Piltant, Olivier (2008), "Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin–Schreier and purely inseparable coverings", `*`F33f`_`[Journal of Algebra`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Journal_of_Algebra]`_`f`*, `!320`! (3): 1051–1082, `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1016/j.jalgebra.2008.03.032, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 2427629

• `:citerefcossartpiltant2009`aCossart, Vincent; Piltant, Olivier (2009), "Resolution of singularities of threefolds in positive characteristic. II" (PDF), `*`F33f`_`[Journal of Algebra`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Journal_of_Algebra]`_`f`*, `!321`! (7): 1836–1976, `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1016/j.jalgebra.2008.11.030, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 2494751

• `:citerefcutkosky2009`aCutkosky, Steven Dale (2009), "Resolution of singularities for 3-folds in positive characteristic", `*Amer. J. Math.`*, `!131`! (1): 59–127, `F33f`_`[arXiv`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ArXiv_(identifier)]`_`f:math/0606530, `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1353/ajm.0.0036, `F33f`_`[JSTOR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=JSTOR_(identifier)]`_`f 40068184, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 2488485, `F33f`_`[S2CID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S2CID_(identifier)]`_`f 2139305

• `:citereftemkin2013`aTemkin, Michael (2013), "Inseparable local uniformization", `*J. Algebra`*, `!373`!: 65–119, `F33f`_`[arXiv`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ArXiv_(identifier)]`_`f:0804.1554, `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1016/j.jalgebra.2012.09.023, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 2995017, `F33f`_`[S2CID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S2CID_(identifier)]`_`f 115167009

• `:citerefzariski1939`a`F33f`_`[Zariski, Oscar`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Oscar_Zariski]`_`f (1939), "The reduction of the singularities of an algebraic surface", `*Ann. of Math.`*, 2, `!40`! (3): 639–689, `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.2307/1968949, `F33f`_`[JSTOR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=JSTOR_(identifier)]`_`f 1968949

• `:citerefzariski1940`aZariski, Oscar (1940), "Local uniformization on algebraic varieties", `*Ann. of Math.`*, 2, `!41`! (4): 852–896, `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.2307/1968864, `F33f`_`[JSTOR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=JSTOR_(identifier)]`_`f 1968864, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 0002864

• `:citerefzariski1944a`a`F33f`_`[Zariski, Oscar`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Oscar_Zariski]`_`f (1944a), "The compactness of the Riemann manifold of an abstract field of algebraic functions", `*`F33f`_`[Bulletin of the American Mathematical Society`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Bulletin_of_the_American_Mathematical_Society]`_`f`*, `!50`! (10): 683–691, `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1090/S0002-9904-1944-08206-2, `F33f`_`[ISSN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISSN_(identifier)]`_`f 0002-9904, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 0011573

• `:citerefzariski1944b`a`F33f`_`[Zariski, Oscar`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Oscar_Zariski]`_`f (1944b), "Reduction of the singularities of algebraic three dimensional varieties", `*Ann. of Math.`*, 2, `!45`! (3): 472–542, `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.2307/1969189, `F33f`_`[JSTOR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=JSTOR_(identifier)]`_`f 1969189, `F33f`_`[MR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MR_(identifier)]`_`f 0011006

>>External links

• "Local uniformization", `*`F33f`_`[Encyclopedia of Mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Encyclopedia_of_Mathematics]`_`f`*, `F33f`_`[EMS Press`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=European_Mathematical_Society]`_`f, 2001 [1994]

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