Terminal valuations and the Nash problem
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.
- Publication:
-
Inventiones Mathematicae
- Pub Date:
- January 2016
- DOI:
- 10.1007/s00222-015-0597-5
- arXiv:
- arXiv:1404.0762
- Bibcode:
- 2016InMat.203..303D
- Keywords:
-
- Mathematics - Algebraic Geometry;
- Primary 14E18;
- Secondary 14E30;
- 14J17
- E-Print:
- v2: 21 pages, minor changes and corrections following the referees' reports. To appear in Invent. Math