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) $V$ 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 [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 $V$ 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 [3]).

References

 [1] O. Zariski, "Local uniformization on algebraic varieties" Ann. of Math. (2) , 41 (1940) pp. 852–896 MR0002864 Zbl 0025.21601 Zbl 66.1327.02 [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) MR0061846 Zbl 0055.38705 [4] O. Zariski, P. Samuel, "Commutative algebra" , 2 , Springer (1975) MR0389876 MR0384768 Zbl 0313.13001

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 $p>0$ resolution of singularities for varieties of dimension 2, and for varieties of dimension 3 provided $p>5$, has been proved by S.S. Abhyankar [a2].