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.

MSC:

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

Zbl 1052.14011
Full Text:

References:

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