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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

Citations:

Zbl 1052.14011
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References:

[1] Alexeev, Complete moduli in the presence of semiabelian group action, Ann. of Math. (2) 155 (3) pp 611– (2002) · Zbl 1052.14017
[2] Ein, Contact loci in arc spaces, Compos. Math. 140 (5) pp 1229– (2004) · Zbl 1060.14004
[3] Ishii, The Nash problem on arc families of singularities, Duke Math. J. 120 (3) pp 601– (2003)
[4] Ishii, The arc space of a toric variety, J. Algebra 278 (2) pp 666– (2004) · Zbl 1073.14066
[5] Ishii, Arcs, valuations and the Nash map, J. Reine Angew. Math. 588 pp 71– (2005) · Zbl 1082.14007
[6] Ishii, The local Nash problem on arc families of singularities, Ann. Inst. Fourier (Grenoble) 56 (4) pp 1207– (2006) · Zbl 1116.14030
[7] Lejeune-Jalabert, Arcs and wedges on sandwiched surface singularities, Amer. J. Math. 121 (6) pp 1191– (1999) · Zbl 0960.14015
[8] M. Morales, The Nash problem on arcs for surface singularities, arXiv:math/0609629v1 [math. AG].
[9] Nash, Arc structure of singularities, Duke Math. J. 81 (1) pp 31– (1995)
[10] P. D. G. Perez, Bijectiveness of the Nash map for quasi-ordinary hypersurface singularities, Int. Math. Res. Not. IMRN Vol. 2007, article ID mm076 (2007).
[11] Plénat, Résolution du problème des arcs de Nash pour les points doubles rationnels Dn, C. R. Math. Acad. Sci. Paris 340 (10) pp 747– (2005) · Zbl 1072.14004
[12] Plénat, A class ofnon-rational surface singularities with bijective Nash map, Bull. Soc. Math. France 134 (3) pp 383– (2006) · Zbl 1119.14007
[13] Plénat, Families of higher dimensional germs with bijective Nash map, Kodai Math. J. 31 (2) pp 199– (2008) · Zbl 1210.14008
[14] Reguera, Families of arcs on rational surface singularities, Manuscripta Math. 88 (3) pp 321– (1995) · Zbl 0867.14012
[15] Reguera, Image of the Nash map in terms of wedges, C. R. Math. Acad. Sci. Paris 338 (5) pp 385– (2004) · Zbl 1044.14032
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