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Arcs, valuations and the Nash map. (English) Zbl 1082.14007

In a celebrated preprint from 1968, J. F. Nash jun. [cf. Duke Math. J. 81, No. 1, 31–38 (1995; Zbl 0880.14010)] introduced the space of arcs on an isolated singularity of a surface \(X\), and he asked if there is a bijection between the irreducible components of the space of arcs on \(X\) and the exceptional components of a minimal resolution of singularities of \(X\). This problem was extended recently by S. Ishii and J. Kollár [Duke Math. J. 120, 601–620 (2003; Zbl 1052.14011)] to any dimension, they also gave a counterexample in dimension 4. But in dimension two there is no counterexample and until recently there were very few positive answers to this problem by A. J. Reguera [Manuscr. Math. 88, 321–333 (1995; Zbl 0867.14012)] for minimal singularities, A. Reguera and M. Lejeune-Jalabert [Rev. Mat. Iberoam. 19, No.2, 581–612 (2003; Zbl 1058.14006)] for sandwich singularities, C. Plénat [Ann. Inst. Fourier 55, No.3, 805–823 (2005; Zbl 1080.14021)] for the singularity \(D_n\) and the reviewer for a very large class of singularities depending only in the dual graph of the exceptional set in the minimal resolution of singularities.
In the paper under review, the author introduces a map from the set of fat arcs to the set of valuations. A fat arc is an arc which does not factor through any proper closed subvarieties. This map is a generalization of the Nash map and the map defined by L. Ein, R. Lazarsfeld and M. Mustata [Compos. Math. 140, No.5, 1229–1244 (2004; Zbl 1060.14004)]. This paper gives an affirmative answer for a non normal toric variety. Another interesting example is the arc determined by a conjugacy class of a finite group \(G\) which gives the quotient variety \(X=\mathbb C^n/G\). The restriction of the map defined by the author onto a subset of these arcs coincides with the McKay correspondence.

MSC:

14B05 Singularities in algebraic geometry
13A18 Valuations and their generalizations for commutative rings
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14C20 Divisors, linear systems, invertible sheaves
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