Zero dimensional arc valuations on smooth varieties

TL;DR

This paper characterizes the maximal irreducible subsets of the arc space of smooth varieties associated with zero-dimensional arc valuations, linking algebraic valuation ideals and geometric infinitely near points.

Contribution

It provides a detailed description of these subsets for transcendence degree zero valuations on smooth varieties, extending to divisorial valuations on surfaces.

Findings

Describes C(v) algebraically via valuation ideals

Describes C(v) geometrically via infinitely near points

C(v) coincides with known subsets for divisorial valuations on surfaces

Abstract

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.