On Nash theory of arc structure of singularities (Q1191380)

Let \(X\) be an algebraic variety over an algebraically closed field \(k\) with singular locus \(\text{Sing}(X)\) and \(S\subseteq X\) a closed subvariety. Let \(T=\text{Spec} k[[t]]\) and \(T_ N=\text{Spec} k[[t]]/t^{N+1}\). An analytic arc is \(\gamma:T\to X\) (resp. \(T_ N\to X)\); \(\gamma\) is called an \(S\text{-arc}\) if \(\gamma(0)\in S\). Using the strong approximation theorem one can prove that an \(N\)-truncated arc can be lifted to an arc if \(N\) is big enough. The coefficients of an \(N\)- truncated \(S\)-arc define a point in some \(k^ M\). The closure of these points (coming from all \(N\)-truncated \(S\)-arcs) is called the Nash variety \(V(X,S,N)\). The paper gives a foundation of the theory of Nash varieties. They may carry a non-reduced structure which turns out to be useful to characterize the smoothness of \(X\) or \(S\) in terms of \(V(X,S,N)\). Each irreducible component of \(V(X,S,N)\) contains a dense open set the points of which correspond to a family of \(N\)-truncated \(S\)-arcs. It is proved that this family considered as family of irreducible curve singularities is equisingular. Using this fact one may use the corresponding invariants (which are constant in an equisingular family) as invariants of the singularities of \(X\).