We study the scheme of formal arcs on a singular algebraic variety and its
images under truncations. We prove a rationality result for the Poincare series
of these images which is an analogue of the rationality of the Poincare series
associated to p-adic points on a p-adic variety. The main tools which are used
are semi-algebraic geometry in spaces of power series and motivic integration
(a notion introduced by M. Kontsevich). In particular we develop the theory of
motivic integration for semi-algebraic sets of formal arcs on singular
algebraic varieties, we prove a change of variable formula for birational
morphisms and we prove a geometric analogue of a result of Oesterle.Comment: Revised Nov. 1997, to appear in Inventiones Mathematicae, 27 page
Abstract. This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and brings new light to some classical invariants.
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