Germs of arcs on singular algebraic varieties and motivic integration

Let k be a ®eld of characteristic zero. We denote by M the Grothendieck ring of algebraic varieties over k (i.e. reduced separated schemes of ®nite type over k). It is the ring generated by symbols ‰SŠ, for S an algebraic variety over k, with the relations ‰SŠ ˆ ‰S0Š if S is isomorphic to S0; ‰SŠ ˆ ‰S n S0Š ‡ ‰S0Š if S0 is closed in S and ‰S S0Š ˆ ‰SŠ‰S0Š. Note that, for S an algebraic variety over k, the mapping S0 7! ‰S0Š from the set of closed subvarieties of S extends uniquely to a mapping W 7! ‰W Š from the set of constructible subsets of S to M, satisfying ‰W [W 0Š ˆ ‰W Š ‡ ‰W 0Š ÿ ‰W \ W 0Š. We set L :ˆ ‰AkŠ and Mloc :ˆ M‰Lÿ1Š. We denote by M‰T Šloc the subring of Mloc‰‰T ŠŠ generated by Mloc‰T Š and the series …1 ÿ LT b†ÿ1 with a in Z and b in Nnf0g. Let X be an algebraic variety over k. We denote by L…X † the scheme of germs of arcs on X . It is a scheme over k and for any ®eld extension k K there is a natural bijection