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<title>Local uniformization</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Local uniformization</span></span>

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<p>In <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, <b>local uniformization</b> is a weak form of <a href="Resolution_of_singularities" title="Resolution of singularities">resolution of singularities</a>, stating that a variety can be desingularized near any <a href="Valuation_(algebra)" title="Valuation (algebra)">valuation</a>, or in other words that the <a href="Zariski%E2%80%93Riemann_space" title="Zariski–Riemann space">Zariski–Riemann space</a> of the array is in some sense non-singular. Local uniformization was introduced by Zariski (<a href="#CITEREFZariski1939">1939</a>, <a href="#CITEREFZariski1940">1940</a>), 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.

</p><p>Local uniformization of a <a href="Algebraic_variety" title="Algebraic variety">variety</a> at a valuation of its <a href="Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">function field</a> means finding a projective model of the variety such that the <a href="Center_(valuation_ring)" class="mw-redirect" title="Center (valuation ring)">center</a> 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. <a href="#CITEREFZariski1944b">Zariski (1944b)</a> 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.)

</p><p><a href="#CITEREFZariski1940">Zariski (1940)</a> proved local uniformization of varieties in any dimension over fields of <a href="Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> 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 (<a href="#CITEREFAbhyankar1956">1956</a>, <a href="#CITEREFAbhyankar1966">1966</a>) 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. <a href="#CITEREFCutkosky2009">Cutkosky (2009)</a> simplified Abhyankar's long proof. Cossart and Piltant (<a href="#CITEREFCossartPiltant2008">2008</a>, <a href="#CITEREFCossartPiltant2009">2009</a>) extended Abhyankar's proof of local uniformization of 3-folds to the remaining characteristics 2, 3, and 5. <a href="#CITEREFTemkin2013">Temkin (2013)</a> showed that it is possible to find a local uniformization of any valuation after taking a <a href="Purely_inseparable_extension" title="Purely inseparable extension">purely inseparable extension</a> of the function field.

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

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFAbhyankar1956" class="citation cs2">Abhyankar, Shreeram (1956), "Local uniformization on algebraic surfaces over ground fields of characteristic <i>p</i>≠0", <i><a href="Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a></i>, Second Series, <b>63</b> (3): <span class="nowrap">491–</span>526, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970014">10.2307/1970014</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970014">1970014</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0078017">0078017</a></cite></li>

<li><cite id="CITEREFAbhyankar1966" class="citation cs2"><a href="S._S._Abhyankar" class="mw-redirect" title="S. S. Abhyankar">Abhyankar, Shreeram S.</a> (1966), <i>Resolution of singularities of embedded algebraic surfaces</i>, Springer Monographs in Mathematics, Acad. Press, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-662-03580-1">10.1007/978-3-662-03580-1</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-63719-2</bdi></cite> (1998 2nd edition)</li>

<li><cite id="CITEREFCossartPiltant2008" class="citation cs2">Cossart, Vincent; Piltant, Olivier (2008), <a rel="nofollow" class="external text" href="https://hal.archives-ouvertes.fr/hal-00139124">"Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin–Schreier and purely inseparable coverings"</a>, <i><a href="Journal_of_Algebra" title="Journal of Algebra">Journal of Algebra</a></i>, <b>320</b> (3): <span class="nowrap">1051–</span>1082, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jalgebra.2008.03.032">10.1016/j.jalgebra.2008.03.032</a></span>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2427629">2427629</a></cite></li>

<li><cite id="CITEREFCossartPiltant2009" class="citation cs2">Cossart, Vincent; Piltant, Olivier (2009), <a rel="nofollow" class="external text" href="https://hal.archives-ouvertes.fr/hal-00139445/file/IIfinalfinal.pdf">"Resolution of singularities of threefolds in positive characteristic. II"</a> <span class="cs1-format">(PDF)</span>, <i><a href="Journal_of_Algebra" title="Journal of Algebra">Journal of Algebra</a></i>, <b>321</b> (7): <span class="nowrap">1836–</span>1976, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jalgebra.2008.11.030">10.1016/j.jalgebra.2008.11.030</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2494751">2494751</a></cite></li>

<li><cite id="CITEREFCutkosky2009" class="citation cs2">Cutkosky, Steven Dale (2009), "Resolution of singularities for 3-folds in positive characteristic", <i>Amer. J. Math.</i>, <b>131</b> (1): <span class="nowrap">59–</span>127, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0606530">math/0606530</a></span>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1353%2Fajm.0.0036">10.1353/ajm.0.0036</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/40068184">40068184</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2488485">2488485</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:2139305">2139305</a></cite></li>

<li><cite id="CITEREFTemkin2013" class="citation cs2">Temkin, Michael (2013), "Inseparable local uniformization", <i>J. Algebra</i>, <b>373</b>: <span class="nowrap">65–</span>119, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/0804.1554">0804.1554</a></span>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jalgebra.2012.09.023">10.1016/j.jalgebra.2012.09.023</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2995017">2995017</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:115167009">115167009</a></cite></li>

<li><cite id="CITEREFZariski1939" class="citation cs2"><a href="Oscar_Zariski" title="Oscar Zariski">Zariski, Oscar</a> (1939), "The reduction of the singularities of an algebraic surface", <i>Ann. of Math.</i>, 2, <b>40</b> (3): <span class="nowrap">639–</span>689, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1968949">10.2307/1968949</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1968949">1968949</a></cite></li>

<li><cite id="CITEREFZariski1940" class="citation cs2">Zariski, Oscar (1940), "Local uniformization on algebraic varieties", <i>Ann. of Math.</i>, 2, <b>41</b> (4): <span class="nowrap">852–</span>896, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1968864">10.2307/1968864</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1968864">1968864</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0002864">0002864</a></cite></li>

<li><cite id="CITEREFZariski1944a" class="citation cs2"><a href="Oscar_Zariski" title="Oscar Zariski">Zariski, Oscar</a> (1944a), "The compactness of the Riemann manifold of an abstract field of algebraic functions", <i><a href="Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a></i>, <b>50</b> (10): <span class="nowrap">683–</span>691, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1944-08206-2">10.1090/S0002-9904-1944-08206-2</a></span>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9904">0002-9904</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0011573">0011573</a></cite></li>

<li><cite id="CITEREFZariski1944b" class="citation cs2"><a href="Oscar_Zariski" title="Oscar Zariski">Zariski, Oscar</a> (1944b), "Reduction of the singularities of algebraic three dimensional varieties", <i>Ann. of Math.</i>, 2, <b>45</b> (3): <span class="nowrap">472–</span>542, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1969189">10.2307/1969189</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1969189">1969189</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0011006">0011006</a></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Local_uniformization">"Local uniformization"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li></ul></div><div><div class="zim-footer">

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