On the Jung method in positive characteristic

Piltant, Olivier

Annales de l'Institut Fourier, Tome 53 (2003) no. 4, pp. 1237-1258.

Soit $\overline{X}$ un germe de surface normale d’anneau local $\overline{R}$ revêtant un germe de surface régulière $X$ d’anneau local $R$ de caractéristique $p>0$. Étant donnée une extension d’anneaux de valuation $W/V$ dominant birationnellement $\overline{R}/R$, nous étudions l’existence d’une nouvelle paire d’anneaux locaux ${\overline{R}}^{\text{'}}/R^{\text{'}}$ dominant birationnellement $\overline{R}/R$, telle que $R^{\text{'}}$ soit régulier et que ${\overline{R}}^{\text{'}}$ n’ait que des singularités toriques. Cette dernière est construite lorsque $W/V$ est sans défaut ou lorsque le degré $[W:V]$ est $p$.

Let $\overline{X}$ be a germ of normal surface with local ring $\overline{R}$ covering a germ of regular surface $X$ with local ring $R$ of characteristic $p>0$. Given an extension of valuation rings $W/V$ birationally dominating $\overline{R}/R$, we study the existence of a new such pair of local rings ${\overline{R}}^{\text{'}}/R^{\text{'}}$ birationally dominating $\overline{R}/R$, such that $R^{\text{'}}$ is regular and ${\overline{R}}^{\text{'}}$ has only toric singularities. This is achieved when $W/V$ is defectless or when $[W:V]$ is equal to $p$

Classification : 13A18, 14E22, 14J17
Keywords: valuations, coverings, resolution of singularities
Mot clés : valuations, revêtements, résolution des singularités

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