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In this paper we explore the generalized Nash problem for arcs on a germ ofsmooth surface: given two prime divisors above its special point, to determinewhether the arc space of one of them is included in the arc space of the otherone. We prove that this problem is combinatorial and we explore its relationwith several notions of adjacency of plane curve singularities.

On the generalized Nash problem for smooth germs and adjacencies of curve singularities