# 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 |

#### 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 $p>0$ 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].

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 MR0199184 Zbl 0122.38603 |

[a2] | S.S. Abhyankar, "Resolution of singularities of arithmetic surfaces" , Harper & Row (1965) MR200272 |

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Local uniformization.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Local_uniformization&oldid=32028