local uniformization
{{Short description|Concept related to resolving singularities in algebraic geometry}}
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 {{harvs|txt|last=Zariski|year1=1939|year2=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. {{harvtxt|Zariski|1944b}} 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.)
{{harvtxt|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. {{harvs|txt|last=Abhyankar|year1=1956|year2=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. {{harvtxt|Cutkosky|2009}} simplified Abhyankar's long proof. {{harvs|txt|last1=Cossart|last2=Piltant|year1=2008|year2=2009}} extended Abhyankar's proof of local uniformization of 3-folds to the remaining characteristics 2, 3, and 5. {{harvtxt|Temkin|2013}} 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
- {{Citation | last1=Abhyankar | first1=Shreeram | title=Local uniformization on algebraic surfaces over ground fields of characteristic p≠0 | jstor=1970014 | doi=10.2307/1970014 | mr=0078017 | year=1956 | journal=Annals of Mathematics |series=Second Series | volume=63 | pages=491–526 | issue=3}}
- {{citation|first= Shreeram S.|last= Abhyankar|authorlink=S. S. Abhyankar|title=Resolution of singularities of embedded algebraic surfaces|series= Springer Monographs in Mathematics|year= 1966|publisher=Acad. Press |isbn=3-540-63719-2|doi=10.1007/978-3-662-03580-1}} (1998 2nd edition)
- {{Citation | last1=Cossart | first1=Vincent | last2=Piltant | first2=Olivier | title=Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin–Schreier and purely inseparable coverings | doi=10.1016/j.jalgebra.2008.03.032 | mr=2427629 | year=2008 | journal=Journal of Algebra | volume=320 | issue=3 | pages=1051–1082| url=https://hal.archives-ouvertes.fr/hal-00139124 | doi-access=free }}
- {{Citation | last1=Cossart | first1=Vincent | last2=Piltant | first2=Olivier | title=Resolution of singularities of threefolds in positive characteristic. II | doi=10.1016/j.jalgebra.2008.11.030 | mr=2494751 | year=2009 | journal=Journal of Algebra | volume=321 | issue=7 | pages=1836–1976| url=https://hal.archives-ouvertes.fr/hal-00139445/file/IIfinalfinal.pdf }}
- {{citation|mr=2488485|last=Cutkosky|first= Steven Dale|title=Resolution of singularities for 3-folds in positive characteristic|journal=Amer. J. Math.|volume= 131 |year=2009|issue= 1|pages= 59–127|jstor=40068184|doi=10.1353/ajm.0.0036|arxiv=math/0606530|s2cid=2139305}}
- {{citation|mr=2995017|last=Temkin|first= Michael|title=Inseparable local uniformization|journal=J. Algebra |volume=373 |year=2013|pages= 65–119|doi=10.1016/j.jalgebra.2012.09.023|arxiv=0804.1554|s2cid=115167009}}
- {{citation|last=Zariski|first= Oscar |authorlink=Oscar Zariski|title=The reduction of the singularities of an algebraic surface|journal= Ann. of Math. |series= 2|volume=40|year=1939| pages= 639–689|doi=10.2307/1968949|issue=3|jstor=1968949}}
- {{citation|mr=0002864|last=Zariski|first= Oscar|title=Local uniformization on algebraic varieties|journal=Ann. of Math. |series= 2 |volume=41|year=1940|issue=4|pages=852–896|jstor=1968864|doi=10.2307/1968864}}
- {{Citation | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | title=The compactness of the Riemann manifold of an abstract field of algebraic functions | doi=10.1090/S0002-9904-1944-08206-2 |mr=0011573 | year=1944a | journal=Bulletin of the American Mathematical Society | issn=0002-9904 | volume=50 | issue=10 | pages=683–691| doi-access=free }}
- {{citation|last=Zariski|first= Oscar |authorlink=Oscar Zariski|title=Reduction of the singularities of algebraic three dimensional varieties|journal= Ann. of Math. |series= 2 |volume=45|year=1944b|pages= 472–542|doi=10.2307/1969189|mr=0011006|issue=3 |jstor=1969189}}