Zero dimensional arc valuations on smooth varieties

For a normalized transcendence degree zero arc valuation v on a nonsingular variety X (with dim X > 1), we describe the maximal irreducible subset C(v) of the arc space of X such that the valuation given by the order of vanishing along a general arc of C(v) equals v. We describe C(v) both algebraically, in terms of the sequence of valuation ideals of v, and geometrically, in terms of the sequence of infinitely near points associated to v. When X is a surface, our construction also applies to any divisorial valuation v, and in this case C(v) coincides with a subset Ein, Lazarsfeld, and Mustata associate to v.