View source for Local uniformization

For a [[Local ring|local ring]], this is the determination of a regular local ring birationally equivalent to it. For an irreducible algebraic variety (cf. [[Irreducible variety|Irreducible variety]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l0602301.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l0602302.png" /> a resolving system is a family of irreducible projective varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l0602303.png" /> birationally equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l0602304.png" /> (that is, such that the rational function fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l0602305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l0602306.png" /> are isomorphic) and satisfying the following condition: For any valuation (place) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l0602307.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l0602308.png" /> there is a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l0602309.png" /> such that the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l06023010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l06023011.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l06023012.png" /> is a non-singular point. The existence of a resolving system (the local uniformization theorem) was proved for arbitrary varieties over a field of characteristic zero (see [[#References|[1]]]), and also for two-dimensional varieties over any field and three-dimensional varieties over an algebraically closed field of characteristic other than 2, 3 or 5 (see [[#References|[2]]]). The existence of a resolving system for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l06023013.png" /> consisting of a single variety implies resolution of the singularities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l06023014.png" /> and can be obtained from the local uniformization theorem in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l06023015.png" />. In the general case the local uniformization theorem implies the existence of a finite resolving system (see [[#References|[3]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  O. Zariski,   "Local uniformization on algebraic varieties"  ''Ann. of Math. (2)'' , '''41'''  (1940)  pp. 852–896</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.S. Abhyankar,   "Resolution of singularities of arithmetic surfaces" , Acad. Press  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  W.V.D. Hodge,   D. Pedoe,   "Methods of algebraic geometry" , '''3''' , Cambridge Univ. Press  (1954)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  O. Zariski,   P. Samuel,   "Commutative algebra" , '''2''' , Springer  (1975)</TD></TR></table>



====Comments====
The resolution of singularities for algebraic varieties of arbitrary dimension over an algebraically closed field of characteristic zero has been achieved by H. Hironaka in 1964 [[#References|[a1]]]. Over algebraically closed fields of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l06023016.png" /> resolution of singularities for varieties of dimension 2, and for varieties of dimension 3 provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l06023017.png" />, has been proved by S.S. Abhyankar [[#References|[a2]]].

For (local) uniformization in analytic geometry and in the theory of functions of a complex variable (Riemann surfaces) cf. [[Uniformization|Uniformization]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Hironaka,   "Resolution of singularities of an algebraic variety over a field of characteristic zero"  ''Ann. of Math.'' , '''79'''  (1964)  pp. 109–326</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.S. Abhyankar,   "Resolution of singularities of arithmetic surfaces" , Harper &amp; Row  (1965)</TD></TR></table>