Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Drinfeld-Grinberg-Kazhdan theorem and embedding codimension of the arc space

Christopher Heng Chiu

TL;DR

This work extends the Drinfeld-Grinberg-Kazhdan theorem to arcs with arbitrary residue fields by constructing a scheme of formal models Z and a deformation-bijection that ties formal neighborhoods of X_infty to finite type data Z, while carefully handling residue-field extensions via pro-objects and CdFD24. It then proves that the embedding codimension ecodim(O_{X_infty, α}) is generically constant on irreducible subsets not contained in Sing X, and that for maximal divisorial sets the ecodim of finite formal models aligns with invariants of singularities such as Mather and Mather-Jacobian discrepancies. The approach combines Weierstrass preparation, deformation theory, and residue-field analysis to relate arc-space local invariants to finite-type schemes, yielding new connections between finite formal models and singularity data. Overall, the results advance understanding of how arc-space invariants reflect singularities and set groundwork for expressing discrepancy-type information via finite formal models and their embeddings into schemes of finite type.

Abstract

We prove an extension of the theorem of Drinfeld, Grinberg and Kazhdan to arcs with arbitrary residue field. As an application we show that the embedding codimension is generically constant on each irreducible subset of the arc space which is not contained in the singular locus. In the case of maximal divisorial sets, this relates the corresponding finite formal models with invariants of singularities of the underlying variety.

The Drinfeld-Grinberg-Kazhdan theorem and embedding codimension of the arc space

TL;DR

This work extends the Drinfeld-Grinberg-Kazhdan theorem to arcs with arbitrary residue fields by constructing a scheme of formal models Z and a deformation-bijection that ties formal neighborhoods of X_infty to finite type data Z, while carefully handling residue-field extensions via pro-objects and CdFD24. It then proves that the embedding codimension ecodim(O_{X_infty, α}) is generically constant on irreducible subsets not contained in Sing X, and that for maximal divisorial sets the ecodim of finite formal models aligns with invariants of singularities such as Mather and Mather-Jacobian discrepancies. The approach combines Weierstrass preparation, deformation theory, and residue-field analysis to relate arc-space local invariants to finite-type schemes, yielding new connections between finite formal models and singularity data. Overall, the results advance understanding of how arc-space invariants reflect singularities and set groundwork for expressing discrepancy-type information via finite formal models and their embeddings into schemes of finite type.

Abstract

We prove an extension of the theorem of Drinfeld, Grinberg and Kazhdan to arcs with arbitrary residue field. As an application we show that the embedding codimension is generically constant on each irreducible subset of the arc space which is not contained in the singular locus. In the case of maximal divisorial sets, this relates the corresponding finite formal models with invariants of singularities of the underlying variety.

Paper Structure

This paper contains 12 sections, 20 theorems, 83 equations.

Table of Contents

  1. Preliminaries
  2. Weierstrass preparation and division theorem
  3. Describing completions via deformations
  4. The arc space of an algebraic variety
  5. On the Drinfeld-Grinberg-Kazhdan theorem
  6. Reduction to the case of complete intersection
  7. The scheme of formal models
  8. A bijection of deformations
  9. Formal neighborhoods via deformations
  10. Application to embedding codimension
  11. On embedding codimension
  12. Embedding codimension over maximal divisorial sets

Key Result

Theorem 1

Let $X$ be a scheme locally of finite type over a perfect field $k$ and $\beta \in X_\infty \setminus (\mathop{\mathrm{Sing}}\nolimits X)_\infty$. Then there exists a locally closed subset $V$ of $X_\infty$ containing $\beta$, a scheme $Z$ of finite type over $k$ and a morphism such that for each $\alpha \in V$ there exists, up to finite separable extension of coefficient fields, an isomorphism b

Theorems & Definitions (49)

  • Theorem 1
  • Theorem 2
  • Definition 1.1
  • Proposition 1.2: Weierstrass preparation and division
  • Lemma 1.3
  • proof
  • Remark 1.4
  • Definition 1.5
  • Remark 1.6
  • Definition 1.7
  • ...and 39 more