Mather Discrepancy as an Embedding Dimension in the Space of Arcs

Abstract

Abstract. Let X be a variety over a field k and let X ∞ be its space of arcs. We study the embedding dimension of the complete local ring A := O X∞,PE where P E is the stable point defined by a divisorial valuation ν E on X. Assuming char k = 0, we prove that embdim A = k E + 1 where k E is the Mather discrepancy of X with respect to ν E . We also obtain that dim A has as lower bound the MatherJacobian log-discrepancy of X with respect to ν E . For X normal and complete intersection, we prove as a consequence that points P E of codimension one in X ∞ have discrepancy k E ≤ 0.

Finite formal model of toric singularities

“…Such a result can be compared to [22], where Reguera and Mourtada have obtained similar results in the computation of the embedding dimension of other types of formal neighborhoods in arc schemes.…”

“…The first breakthrough in this way has been obtained by Reguera in [25]. The present work can be linked, in spirit, to the subsequent work [22].…”

“…Then the minimal finite formal model S n is of dimension (n) − 1 and embedding dimensionK n .By theorem 2.7 and example 6.2, the conclusion holds in particular if n is strongly essential.Remark 6.5. Let us note that theorem 6.3 presents analogies with the main result of[22].Proof. (of theorem 6.3) Recall thatỸ n (resp.…”

Differentials on the arc space

“…These and related invariants have been studied in the literature (e.g., see [ELM04,dFEI08,dFM15,Reg,MR]). Both numbers provide measures of the "size" of the point.…”

“…By definition, these are the generic points of the irreducible constructible subsets of X ∞ (see Section 11 for a discussion of the notion of constructibility in arc spaces). Stable points and their local rings have been extensively studied in [Reg06,Reg09,Reg,MR]. The following theorem can be viewed as providing a characterization of stable points that are not fully contained, as arcs, within the singular locus of X.…”