Some numerical criteria for the Nash problem on arcs for surfaces. (English) Zbl 1178.14004

The Nash problem asks the bijectiveness of the Nash map from the set of families of arcs with the center in the singular locus of a variety, to the set of essential divisors over the singular locus of the variety. This problem was posed by Nash in 1968 and negatively answered in 2003 by J. Kollár and the reviewer [Duke Math. J. 120, No. 3, 601–620 (2003; Zbl 1052.14011)] for dimension higher than 3. The problem is still open for 2 and 3 dimensional cases. This paper shows some 2 dimensional singularities for which the Nash problem is affirmatively answered. The author generalized the result of [C. Plenat and P. Popescu-Pampur, Bull. Soc. Math. France, Bull. Soc. Math. Fr. 134, No. 3, 383–394 (2006; Zbl 1119.14007)] which showed a sufficient condition of the intersection matrix of the minimal resolution of a surface singularity for the affirmative answer to the Nash problem. The condition of this paper for the intersection matrix is not so simple as their condition, but the author obtains many affirmative examples which are not obtained by Plénat and Popescu-Pampu’s result.


14B05 Singularities in algebraic geometry
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