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.


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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