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Image of the Nash map in terms of wedges. (English. Abridged French version) Zbl 1044.14032
Let \(k\) be an uncountable algebraically closed field of characteristic zero and \(X\) a \(k\)-variety with singular locus \(S\). Let \(\pi :X_\infty \rightarrow X\) be the canonical projection from the space of arcs \(X_\infty\) on \(X\) to \(X\). The Nash map is a canonical map \(\mathcal{N}\) from the set of irreducible components of \(X_\infty^S=\pi^{-1}(S)\) into the set of essential components on a resolution of singularities \(Y\) of \(X\). An essential component on \(Y\) is the center on \(Y\) of an essential divisor over \(X\). An essential divisor over \(X\) is a divisorial valuation \(\nu\) of the function field \(k(X)\) of \(X\) centered in \(S\) such that the center \(\nu\) on any desingularization \(p:Y \rightarrow X \) is an irreducible component of the exceptional locus \(p^{-1}(S)\) on \(Y\). The map \(\mathcal{N}\) does not depend on \(Y\). Necessary and sufficient conditions for an essential divisor are given to be in the image of the Nash map. Especially it follows that the Nash problem is true for sandwiched surface singularities.

14P20 Nash functions and manifolds
32S45 Modifications; resolution of singularities (complex-analytic aspects)
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14J17 Singularities of surfaces or higher-dimensional varieties
Full Text: DOI
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