View source for Local uniformization

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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]]) $V$ over a field $k$ a resolving system is a family of irreducible projective varieties $\{V_\alpha\}$ birationally equivalent to $V$ (that is, such that the rational function fields $k(V_\alpha)$ and $k(V)$ are isomorphic) and satisfying the following condition: For any valuation (place) $v$ of $k(V)$ there is a variety $V'\in\{V_\alpha\}$ such that the centre $P'$ of $v$ on $V'$ 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 $V$ consisting of a single variety implies resolution of the singularities of $V$ and can be obtained from the local uniformization theorem in dimension $\leq3$. 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 {{MR|0002864}} {{ZBL|0025.21601}} {{ZBL|66.1327.02}} </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) {{MR|0061846}} {{ZBL|0055.38705}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''2''' , Springer (1975) {{MR|0389876}} {{MR|0384768}} {{ZBL|0313.13001}} </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 $p>0$ resolution of singularities for varieties of dimension 2, and for varieties of dimension 3 provided $p>5$, 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 {{MR|0199184}} {{ZBL|0122.38603}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S.S. Abhyankar, "Resolution of singularities of arithmetic surfaces" , Harper &amp; Row (1965) {{MR|200272}} {{ZBL|}} </TD></TR></table>