Abhyankar places admit local uniformization in any characteristic

Details Abhyankar places admit local uniformization in any characteristic Abhyankar places admit local uniformization in any characteristic

2003-04-12

arxiv.org/abs/math/0304159v1

Mathematics

Algebraic Geometry

21 pages, submitted

Scientific paper

We prove that every place $P$ of an algebraic function field $F|K$ of arbitrary characteristic admits local uniformization, provided that the sum of the rational rank of its value group and the transcendence degree of its residue field $FP$ over $K$ is equal to the transcendence degree of $F|K$, and the extension $FP|K$ is separable. We generalize this result to the case where $P$ dominates a regular local Nagata ring $R\subseteq K$ of Krull dimension $\dim R\leq 2$, assuming that the valued field $(K,v_P)$ is defectless, the factor group $v_P F/v_P K$ is torsion-free and the extension of residue fields $FP|KP$ is separable. The results also include a form of monomialization. Further, we show that in both cases, finitely many Abhyankar places admit simultaneous local uniformization on an affine scheme if they have value groups isomorphic over $v_P K$.

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