Terminal valuations and the Nash problem

By Tommaso de Fernex, Roi Docampo. 2014

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Let $$X$$X be an algebraic variety of characteristic zero. Terminal valuations are defined in the sense of the minimal model program, as those valuations given by the exceptional divisors on a minimal model over $$X$$X. We prove that every terminal valuation over $$X$$X is in the image of the Nash map, and thus it corresponds to a maximal family of arcs through the singular locus of $$X$$X. In dimension two, this result gives a new proof of the theorem of Fernández de Bobadilla and Pe Pereira stating that, for surfaces, the Nash map is a bijection.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b4a220e3758e2ccb613acc356a7f78f4777d04e5