# Terminal valuations and the Nash problem

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## Abstract

Let \(X\) 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 \(X\). We prove that every terminal valuation over \(X\) is in the image of the Nash map, and thus it corresponds to a maximal family of arcs through the singular locus of \(X\). 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.

## Mathematics Subject Classification

Primary 14E18 Secondary 14E30 14J17## Notes

### Acknowledgments

We are very grateful to Javier Fernández de Bobadilla for many conversations and for bringing to our attention an error in an earlier version of this paper. We thank Lawrence Ein and Shihoko Ishii for useful discussions, and the referees for their careful reading of the paper and their relevant comments.

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