Arc valuations on smooth varieties

Abstract

Let X be a nonsingular variety (with dim X ⩾ 2 ) over an algebraically closed field k of characteristic zero. Let α : Spec k 〚 t 〛 → X be an arc on X, and let v = ord α be the valuation given by the order of vanishing along α. We describe the maximal irreducible subset C ( v ) of the arc space of X such that val C ( v ) = 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. As a corollary, we get that v is determined by its sequence of centers. Also, when X is a surface, our construction also applies to any divisorial valuation v, and in this case C ( v ) coincides with the one introduced in [L. Ein, R. Lazarsfeld, M. Mustaţǎ, Contact loci in arc spaces, Compos. Math. 140 (2004) 1229–1244, Example 2.5].