Germs of arcs on singular algebraic varieties and motivic integration

Details Germs of arcs on singular algebraic varieties and motivic integration Germs of arcs on singular algebraic varieties and motivic integration

1998-03-11

arxiv.org/abs/math/9803039v1

Invent. Math. 135 (1999), no. 1, 201--232

Mathematics

Algebraic Geometry

Revised Nov. 1997, to appear in Inventiones Mathematicae, 27 pages

Scientific paper

10.1007/s002220050284

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.

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