Inseparable local uniformization
The paper under review deals with local uniformization of algebraic varieties, which can be considered as the local part of desingularization. The author proves that it always hold after a purely inseparable extension of the function field of the variety. More precisely, let \(X\) be an integral algebraic variety. Then \(X\) may be covered by a finite number of integral and regular varieties \(Y_{1},\dotsc,Y_{n}\) such that the extensions \(k(Y_{i})/k(X)\) are all purely inseparable. The proof goes by induction on the dimension and involves fibrating by curves. It is worth noticing that the proof of the induction step relies on inseparable local uniformization for some points on non-Archimedean \(k\)-analytic curves (type 1 and 4 in Berkovich's terminology; they are called terminal by the author). A decompletion procedure then allows to deduce a uniformization result for certains valuations on a curve over \(k^\circ\). The base case of the induction, which consists in local uniformization for Abhyankar valuations, is proved at the end of the paper. It involves techniques from logarithmic geometry.
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