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

Technical Summary: The Drinfeld–Grinberg–Kazhdan Theorem and Embedding Codimension of the Arc Space

Problem Statement
The geometry of the arc space $X_\infty$ of an algebraic variety $X$ encodes significant information regarding the singularities of $X$. A central object of study is the maximal divisorial set $C_\nu(X)$, the closure of arcs inducing a specific divisorial valuation $\nu$. While the Drinfeld–Grinberg–Kazhdan (DGK) theorem establishes that for a $k$-rational arc $\alpha$ not contained in the singular locus, the formal neighborhood $(X_\infty, \alpha)$ is isomorphic to the formal neighborhood of a finite type scheme $Z$ (a "finite formal model") times a formal affine space, this result was previously limited to $k$-rational points over perfect fields.

A major open question concerns the relationship between these finite formal models and the invariants of the underlying singularity. Specifically, it is unclear how the formal models of general arcs (those with arbitrary residue fields) relate to the formal model of the generic arc $\alpha_\nu$ of a maximal divisorial set. Furthermore, the behavior of the embedding codimension, $ecodim(\mathcal{O}_{X_\infty, \alpha})$, across irreducible subsets of the arc space requires clarification, particularly regarding its constancy and its relation to the smoothness of $X$.

Methodology
The paper employs a strategy of generalizing the DGK theorem to arcs with arbitrary residue fields, thereby avoiding the explicit construction of coefficient fields which often obstructs generalization.

1. Generalization of the DGK Theorem: The author extends the proof of Drinfeld [14] to arcs $\beta \in X_\infty \setminus (\text{Sing } X)_\infty$ with arbitrary residue fields. The proof follows Drinfeld's deformation-theoretic approach:

   • Reduction to Complete Intersections: The problem is reduced to the case where $X$ is a complete intersection via an open neighborhood argument involving the Jacobian minor.
   • Scheme of Formal Models: A scheme $Z$ of finite type is constructed to parametrize the "finite formal models." This involves defining a scheme $W$ of polynomials and imposing conditions derived from the Weierstrass preparation and division theorems.
   • Deformation Bijection: A natural bijection is established between the functors of deformations of the arc space at $\alpha$ and the scheme $Z \times \mathbb{A}^\infty$ at the image point $\mu(\alpha)$.
   • Residue Field Extensions: To handle non-$k$-rational arcs, the author utilizes results from [12] regarding finite separable extensions of residue fields induced by linear projections. This allows the construction of an isomorphism between formal neighborhoods up to a finite separable extension of coefficient fields, without explicitly constructing the coefficient field itself.

2. Embedding Codimension Analysis: The paper defines embedding codimension for non-Noetherian rings (specifically quasi-adic rings in the category of pro-objects) and investigates its behavior.

   • Upper Semicontinuity: It is shown that for schemes of finite type, the embedding codimension is not generally upper semicontinuous, but it becomes constant on a nonempty open subset of any irreducible closed subset not contained in the singular locus.
   • Application to Maximal Divisorial Sets: The generalized DGK theorem is applied to maximal divisorial sets $C_\nu(X)$ to relate the embedding codimension of general arcs in $C_\nu(X)$ to that of the generic arc $\alpha_\nu$.

3. Characterization of Smoothness: The paper extends a theorem by Bourqui and Sebag [6] characterizing arcs with embedding codimension 0. By combining the constancy of embedding codimension with the existence of rational points (guaranteed by uncountability of the field or the stability of the arc), the author relates the vanishing of embedding codimension to the smoothness of the base variety.

Key Contributions and Results

• Theorem A (Generalized DGK Theorem): For a scheme $X$ locally of finite type over a perfect field $k$ and an arc $\beta \notin (\text{Sing } X)_\infty$, there exists a locally closed subscheme $V$ containing $\beta$, a finite type scheme $Z$, and a morphism $\mu: V \to Z \times \mathbb{A}^N$ such that for every $\alpha \in V$, the formal neighborhood $(X_\infty, \alpha)$ is isomorphic to $(Z \times \mathbb{A}^N, \mu(\alpha))$ up to a finite separable extension of coefficient fields. This fully recovers the original DGK theorem for $k$-rational points.
• Theorem B (Constancy of Embedding Codimension): For any irreducible closed subset $W \subset X_\infty$ not contained in $(\text{Sing } X)_\infty$, the function $\alpha \mapsto ecodim(\mathcal{O}_{X_\infty, \alpha})$ is finite and constant on a nonempty open subset of $W$. Consequently, for any maximal divisorial set $C_\nu(X)$, the embedding codimension of a finite formal model for a general $k$-rational arc equals that of the local ring at the maximal divisorial arc $\alpha_\nu$.
• Theorem C (Characterization of Smoothness): Let $k$ be a perfect field, $X$ a variety, and $\alpha \in X_\infty$. Assuming either $k$ is uncountable or $\alpha$ is stable, if $X$ is geometrically unibranch at $\alpha(0)$, then $ecodim(\mathcal{O}_{X_\infty, \alpha}) = 0$ if and only if $X$ is smooth at $\alpha(0)$. This generalizes the result of Bourqui and Sebag to non-$k$-rational arcs under specific conditions.

Significance and Limitations
The paper provides a rigorous framework for studying the local geometry of arc spaces at points with arbitrary residue fields, removing the restriction to $k$-rational points that characterized previous work. By establishing the constancy of embedding codimension on maximal divisorial sets, it offers a step toward understanding the precise relation between finite formal models and singularity invariants (such as Mather discrepancies).

However, the author explicitly notes that the relationship between the formal models of general arcs and the generic arc remains subtle. While the embedding codimension is constant, the dimension and embedding dimension of the minimal formal models may differ between general arcs and the generic arc (as observed in prior work on curves). The paper raises Question 3.18: whether the embedding dimension of the minimal formal model of a general $k$-rational arc in $C_\nu(X)$ equals that of the minimal formal model of the generic arc $\alpha_\nu$. The author suggests that while Theorem A provides the necessary tools, a precise answer is not yet obvious.

Furthermore, the paper addresses Reguera's Question 2.10 regarding the regularity of the local ring at a stable arc. Theorem 3.23 provides a positive answer under the stronger assumption that $X$ is geometrically unibranch. The paper demonstrates via Proposition 3.24 that if $X$ is merely unibranch (but not geometrically unibranch), the local ring $\mathcal{O}_{X_\infty, \alpha}$ may have embedding codimension 0 without being regular (or even reduced), indicating that Reguera's original question remains open for the case of multiple geometric branches.