Local uniformization

For a local ring, this is the determination of a regular local ring birationally equivalent to it. For an irreducible algebraic variety (cf. Irreducible variety) over a field a resolving system is a family of irreducible projective varieties birationally equivalent to (that is, such that the rational function fields and are isomorphic) and satisfying the following condition: For any valuation (place) of there is a variety such that the centre of on 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 [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 [2]). The existence of a resolving system for consisting of a single variety implies resolution of the singularities of and can be obtained from the local uniformization theorem in dimension . In the general case the local uniformization theorem implies the existence of a finite resolving system (see [3]).

References

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

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 [a1]. Over algebraically closed fields of characteristic resolution of singularities for varieties of dimension 2, and for varieties of dimension 3 provided , has been proved by S.S. Abhyankar [a2].

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

References

[a1]	H. Hironaka, "Resolution of singularities of an algebraic variety over a field of characteristic zero" Ann. of Math. , 79 (1964) pp. 109–326
[a2]	S.S. Abhyankar, "Resolution of singularities of arithmetic surfaces" , Harper & Row (1965)