Jet schemes and singularities. (English) Zbl 1181.14019

Ein, Lawrence; Mustaţă, Mircea

Abramovich, D. (ed.) et al., Algebraic geometry, Seattle 2005. Proceedings of the 2005 Summer Research Institute, Seattle, WA, USA, July 25–August 12, 2005. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4703-9/hbk; 978-0-8218-4057-3/set). Proceedings of Symposia in Pure Mathematics. 80, Pt. 2, 505-546 (2009).

The goal of this paper is to give a proof of a version of inversion of adjunction by means of jet schemes and arc space. The authors present a self contained exposition for this aim. First they introduce the notion of jet schemes and show their existence and the basic properties. They also give the description on the structure of the fibers of the truncation morphisms. They show the formula of minimal log discrepancy in terms of the codimension of cylinders in the arc space. Finally, they prove the inversion of adjunction; i.e., the equality of minimal log discrepancies of a pair and the restriction of the pair onto the normal subvariety.
For the entire collection see [Zbl 1158.14004].

Reviewer: Shihoko Ishii (Tokyo)

MSC:

14E30	Minimal model program (Mori theory, extremal rays)
14B05	Singularities in algebraic geometry
14E99	Birational geometry
14J10	Families, moduli, classification: algebraic theory