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The Nash problem on arc families of singularities. (English) Zbl 1052.14011
Summary: J. F. Nash jun. [Duke Math. J. 81, No. 1, 31–38 (1995; Zbl 0880.14010)] proved that every irreducible component of the space of arcs through a singularity corresponds to an exceptional divisor that appears on every resolution. He asked if the converse also holds: Does every such exceptional divisor correspond to an arc family? We prove that the converse holds for toric singularities but fails in general.

MSC:
14C20 Divisors, linear systems, invertible sheaves
14B05 Singularities in algebraic geometry
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14J10 Families, moduli, classification: algebraic theory
14J17 Singularities of surfaces or higher-dimensional varieties
14B20 Formal neighborhoods in algebraic geometry
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