## Bijectiveness of the Nash map for quasi-ordinary hypersurface singularities.(English)Zbl 1129.14004

The Nash problem (resp. local Nash problem) asks the bijectiveness of the Nash map (resp. local Nash map) from the set of families of arcs with the origin in the singular locus of a variety (resp. at a fixed singular point), to the set of essential divisors over the singular locus of the variety (resp. over the point). In [S. Ishii, Ann. Inst. Fourier 56, No. 6, 1207–1224 (2006; Zbl 1116.14030)] the bijectiveness of the local Nash map for a quasi-ordinary hypersurface singularity was proved. The reason why only the local version was proved is because the definition of a quasi-ordinary singularities is about the completion of the local ring at the singular point and one can not see the situation of an “algebraic” neighborhood (Zariski open neighborhood in the algebraic variety) of the singularity. But in this paper, the author considers the spectrum of the completion a the local ring of the singularity (i.e. algebro analytic neighborhood of the singularity) instead of an algebraic neighborhood of the singularity. He succeeds to prove the bijectiveness of the Nash map on this scheme, the spectrum of the complete local ring.

### MSC:

 14B05 Singularities in algebraic geometry 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14Q10 Computational aspects of algebraic surfaces

### Keywords:

arc spaces; Nash problem

Zbl 1116.14030
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