Arc valuations on smooth varieties

JOURNAL OF ALGEBRA

卷:321

Article

More, Yogesh

关键词: Valuation theory;    Arc spaces;   

DOI  :  10.1016/j.jalgebra.2009.02.002

来源: Elsevier

【 摘 要 】

Let X be a nonsingular variety (with dim X >= 2) over an algebraically closed field k of characteristic zero. Let alpha :Spec k[t] -> X be an arc on X, and let v = ord(alpha) be the valuation given by the order of vanishing along alpha. 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. Lazars-feld, M. Mustata, Contact loci in arc spaces, Compos. Math. 140 (2004) 1229-1244, Example 2.5]. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

Free