Nash problem for stable toric varieties. (English) Zbl 1182.14008

The space of arcs \(X_\infty \) of an algebraic variety \(X\) encodes important information about the geometry of \(X\). Let \(\pi :X_\infty\rightarrow X\), be the canonical map, then \(\pi ^{-1}(\mathrm{Sing}(X))\) is a union of irreducible components. An irreducible component \(C\) is called Nash component if it contains an arc \(\alpha \) such that \(\alpha (\eta )\not\in \mathrm{Sing}(X)\), where \(\eta \) is the generic point of \(\mathrm{Spec} k[[t]]\). Nash has proved that there is an injective map from the irreducible components of \(\pi ^{-1}(\mathrm{Sing}(X))\) to the set of essential divisors over \(X\), and ask if this map is bijective. The problem is not completely solved in dimension two, but there are some important contributions by Monique Lejeune-Jalabert, A. Reguera, Camille Plenat and the reviewer. S. Ishii and J. Kollár [Duke Math. J. 120, 601–620 (2003; Zbl 1052.14011)] proved it for toric varieties in any dimension.
In this paper the author gives a partial answer to this problem in the case of a stable toric variety. In fact the author studies the Nash problem for pairs \((X,Y)\), where \(X\) is an algebraic variety and \(Y\subset X\) is a proper non empty closed subset.


14E18 Arcs and motivic integration
14B05 Singularities in algebraic geometry


Zbl 1052.14011
Full Text: DOI arXiv


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