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Detecting and Quantifying Isolated Singularities over Discrete Valuation Rings

This paper establishes a theory of isolated hypersurface singularities over discrete valuation rings in mixed characteristic by introducing analogues of Tjurina and Milnor numbers, proving generalized determinacy and Mather-Yau theorems, and defining numerical invariants to distinguish between unramified and ramified cases.

Original authors: Yotam Svoray

Published 2026-03-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yotam Svoray

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to understand the stability of a building. In the world of mathematics, these "buildings" are shapes defined by equations, and the "stability" refers to whether the shape has any sharp, jagged points called singularities.

For a long time, mathematicians could only study these buildings when they were made of "perfect" materials (like complex numbers or fields). But in the real world of advanced math, we often have to deal with "mixed" materials—structures built over Discrete Valuation Rings (DVRs). Think of a DVR as a building site where the ground isn't perfectly flat; it has a specific "grain" or direction (represented by a special element called $\pi$ , the uniformizer) that behaves differently than the other materials.

Yotam Svoray's paper is like a new manual for architects who want to inspect these mixed-material buildings for cracks and weak spots. Here is the breakdown of his work using simple analogies:

1. The Problem: The "Grain" of the Wood

In standard math (over fields), if you have a shape defined by $x^2 + y^2$ , you can rotate or stretch it easily to see if it's a "smooth" circle or a "pointy" cusp.

However, in this mixed world, there is a special ingredient, $\pi$ (think of it as a specific type of wood grain or a heavy metal beam).

The Issue: You can't just rotate the building freely. If you try to swap a piece of "grain" ( $\pi$ ) with a regular piece of wood ( $x$ ), the structure changes completely.
The Example: Imagine two houses. One has a foundation made of $x^2 + \pi^3$ , the other $x^3 + \pi^2$ . In a normal world, you might think they are just different angles of the same house. But here, because $\pi$ is "heavy" and special, they are fundamentally different. One might have a sharp corner (a cusp), and the other a smooth bump. You can't just "change variables" to make them look the same.

2. The Solution: The "Ghost Variable" Trick

To solve this, Svoray introduces a clever trick. He says: "Let's pretend $\pi$ is just another variable, like $y$ ."

The Metaphor: Imagine you are looking at a painting where one color (let's say "Gold") is special and behaves differently than "Red" or "Blue." To analyze the painting, you temporarily treat "Gold" as if it were just another color, "Yellow."
The Result: By turning $\pi$ into a "Ghost Variable" ( $y$ ), Svoray can use all the powerful tools mathematicians already have for analyzing smooth shapes. He creates a "shadow" of the equation where $\pi$ is just a regular variable. He calls this shadow $\tilde{f}$ .

3. The Tools: New "Rulers" for Measuring Cracks

Mathematicians usually use two rulers to measure how bad a singularity is: the Milnor Number and the Tjurina Number.

Old Rulers: These worked great for fields but broke when applied to these mixed-material rings.
Svoray's New Rulers:
1. $\tau_V(f)$ : This is the "Ghost Ruler." It measures the complexity of the shadow ( $\tilde{f}$ ). It tells you if the shape is generally smooth, but it's not perfect for detecting every type of crack in this specific mixed environment.
2. $\tau(f, \delta)$ and $\tau^\Delta(f)$ : These are "Specialized Rulers" for Unramified rings (where the ground is stable). They use a concept called a " $p$ -derivation," which is like a special scanner that looks at how the shape changes specifically with respect to the prime number $p$ .
3. $\tau^\pi(f)$ : This is the ruler for Ramified rings (where the ground is unstable). It uses a "derivative with respect to $\pi$ ," essentially asking, "How does the shape change if I wiggle the special grain?"

4. The Big Discoveries

Svoray proves several "Theorems" that act as safety checks for these buildings:

The "Finite Determinacy" Theorem: This says that to know if a building is stable, you don't need to check every single brick. You only need to check the first few layers of bricks (a finite number). If the first few layers are okay, the whole building is safe. This is huge because it turns an infinite problem into a finite one.
The "Mather-Yau" Theorem: This is a way to tell if two buildings are actually the same shape, just painted differently. Svoray proves that if you look at the "blueprints" of the buildings (specifically the ideal generated by the function and its derivatives), you can tell if they are equivalent.
Detecting the "Isolated" Singularity: The paper gives clear criteria to say, "Yes, this building has exactly one weak spot, and it's right here."
- If the ring is Unramified, you check if the "Ghost Ruler" and the "Special Scanner" give finite numbers.
- If the ring is Ramified, you check the "Derivative with respect to $\pi$ " ruler.

5. Why Does This Matter?

Think of this as upgrading the safety code for skyscrapers.

Before, engineers (mathematicians) only knew how to check skyscrapers built on perfect concrete (fields).
Now, Svoray has given them the tools to check skyscrapers built on tricky, mixed terrain (mixed characteristic).
This allows mathematicians to classify and understand complex shapes that were previously too messy to analyze, bridging the gap between the "perfect" world of fields and the "messy" reality of number theory and algebraic geometry.

In a nutshell: Svoray figured out how to treat a stubborn, special ingredient ( $\pi$ ) as a regular variable to use old tools, and then invented new, specialized tools to measure exactly how "broken" a shape is when that special ingredient is involved. He proved that these new tools work just as reliably as the old ones, even in the most complex mathematical environments.

1. Problem Statement

The paper addresses the challenge of extending the classical theory of isolated hypersurface singularities from algebraically closed fields (characteristic 0 or $p$ ) to the setting of mixed characteristic $(0, p)$ . Specifically, the author investigates quotient rings of the form $V[[x]]/\langle f \rangle$ , where $V$ is a Discrete Valuation Ring (DVR) with uniformizer $\pi$ .

Key Challenges:

Non-invertibility of $\pi$ : Unlike fields, the uniformizer $\pi$ in a DVR is not invertible. This restricts the group of $V$ -automorphisms of the power series ring $V[[x]]$ , making standard notions of "equivalence up to change of variables" insufficient. For instance, $x_1^2 + x_2^2 + \pi^3$ and $x_1^3 + x_2^2 + \pi^2$ are not isomorphic over $V[[x]]$ , even though they might appear similar, because their reductions modulo $\pi$ yield different singularity types (cusp vs. ordinary double point).
Defining Invariants: Classical invariants like the Milnor number ( $\mu$ ) and Tjurina number ( $\tau$ ) rely on the Jacobian ideal. In mixed characteristic, the behavior of derivatives changes (e.g., $\partial(x^p)/\partial x = 0$ ), and the standard Jacobian criterion fails to detect singularities correctly in all cases.
Formalizing "Variable" Treatment: There is a need to treat the uniformizer $\pi$ formally as a variable to apply techniques from characteristic 0, but doing so requires a rigorous definition that respects the unique expansion of elements in a DVR.

2. Methodology

The author develops a framework that treats the uniformizer $\pi$ as an additional formal variable $y$ .

The "Dummy Variable" Construction (Notation 3.1):
Every element $f \in V[[x]]$ can be uniquely written as $f = \sum u_\alpha \pi^{n_\alpha} x^\alpha$ , where $u_\alpha$ are units. The author defines a map $f \mapsto \tilde{f}$ where $\tilde{f}(x, y) = \sum u_\alpha y^{n_\alpha} x^\alpha \in V[[x, y]]$ . This effectively replaces $\pi$ with a formal variable $y$ , allowing the use of standard power series techniques in one higher dimension.
Equivalence Relations:
Two functions $f, g \in V[[x]]$ are defined as equivalent ( $f \sim g$ ) if their lifts $\tilde{f}$ and $\tilde{g}$ are contact equivalent in $V[[x, y]]$ (i.e., isomorphic as $V$ -algebras).
Generalized Determinacy:
The paper extends the Determinacy Theorem to complete Noetherian local rings. It establishes conditions under which a function is determined by its finite jet, utilizing the ideal $I$ and the Jacobian ideal $J(f)$ .
Derivations in Mixed Characteristic:
- Unramified Case: Utilizes $p$ -derivations (Buium-Joyal derivations), which act as "derivatives with respect to $p$ ."
- Ramified Case: Utilizes a specific derivation $\frac{\partial}{\partial \pi}$ that acts as a derivative with respect to the uniformizer.

3. Key Contributions and Results

A. Generalized Determinacy and Mather-Yau Theorem (Section 2)

Finite Determinacy Theorem (Theorem 2.4): Proved for any complete Noetherian local ring $R$ . If $f \in I^2$ satisfies $I^{k+2} \subset I\langle f \rangle + I^2 J(f)$ , then $f$ is $(2k - \text{ord}_I(f) + 2)$ -determined with respect to $I$ .
Generalized Mather-Yau Theorem (Proposition 2.33): For finitely determined elements $f, g$ , they are equivalent ( $f \sim_I g$ ) if and only if the truncated algebras $R[[x]]/\langle f, I^k J(f) \rangle$ and $R[[x]]/\langle g, I^k J(g) \rangle$ are isomorphic for all sufficiently large $k$ .
Jacobian Number: Introduced $j_I(f) = \text{length}_R(I / (\langle f \rangle + J(f)))$ as a quantitative measure of determinacy.

B. Tjurina and Milnor Numbers over DVRs (Section 3)

Definition via Lifts: The author defines the Tjurina number $\tau_V(f)$ and Milnor number $\mu_V(f)$ based on the length of the quotient of the lifted ring $V[[x, y]]$ by the ideal generated by $\tilde{f}$ and its Jacobian (intersected with a generic hypersurface to ensure finite length).
Splitting Lemma (Proposition 3.22): Proved a mixed-characteristic version of the Splitting Lemma. If $f$ has order 2 and the residue field is quadratically closed, $f$ is equivalent to $\pi^2 + x_1^2 + \dots + x_k^2 + g(x_{k+1}, \dots, x_n)$ , where $g$ is a residual part of higher order.
Morse's Lemma (Proposition 3.27): Established that $\tau_V(f) = 1$ if and only if $f$ has full rank ( $k=n$ ) and is equivalent to $\pi^2 + x_1^2 + \dots + x_n^2$ .

C. Detecting Isolated Singularities (Section 4)

The paper distinguishes between unramified and ramified DVRs, providing distinct criteria for detecting isolated singularities.

1. Unramified Case ( $V$ has uniformizer $p$ ):

$p$ -Derivations: Uses the operator $\delta$ (a $p$ -derivation) to define a new Tjurina number $\tau(f, \delta)$ .
Detection Criterion (Proposition 4.14): $f$ defines an isolated singularity if and only if $p \in \sqrt{J(f)}$ and $\tau(f, \delta) < \infty$ for any $p$ -derivation $\delta$ .
Invariant $\tau^\Delta(f)$ : Defined as $\tau(f, \delta)$ modulo $p$ . This value is independent of the choice of $\delta$ and provides a robust detection criterion (Corollary 4.22).
Limitation: The value of $\tau(f, \delta)$ depends on the choice of $\delta$ , and these invariants are not preserved under the equivalence relation $\sim$ .

2. Ramified Case ( $p \in \langle \pi^2 \rangle$ ):

Derivation $\partial_\pi$ : Uses a derivation with respect to the uniformizer $\pi$ .
Ramified Tjurina Number (Definition 4.25): $\tau^\pi(f) = \dim_\kappa (\kappa[[x]] / \langle f, J_\pi(f) \rangle)$ , where $J_\pi(f)$ includes $\partial_\pi(f)$ .
Detection Criterion (Proposition 4.26): $f$ defines an isolated singularity if and only if $\pi \in \sqrt{J(f)}$ and $\tau^\pi(f) < \infty$ .
Comparison: $\tau^\pi(f)$ is shown to be an upper bound for the standard $\tau_V(f)$ (Proposition 4.27).

4. Significance and Implications

Unification of Theory: The paper successfully bridges the gap between singularity theory over fields and over mixed-characteristic rings, providing a coherent framework for the latter.
New Invariants: It introduces four distinct numerical invariants ( $\tau_V, \tau(f, \delta), \tau^\Delta, \tau^\pi$ ) tailored to different aspects of the ring structure (unramified vs. ramified, dependence on derivation choice).
Algorithmic Detection: The criteria provided (e.g., checking if $p \in \sqrt{J(f)}$ and finiteness of specific lengths) offer concrete, computable methods to determine if a hypersurface over a DVR has an isolated singularity.
Limitations and Nuances: The work highlights that unlike the field case, "equivalence" in mixed characteristic is more rigid. Invariants like $\tau(f, \delta)$ and $\tau^\pi(f)$ are not invariant under the equivalence relation $\sim$ , revealing a fundamental difference in the geometry of singularities in mixed characteristic.

Conclusion

Yotam Svoray's work provides a foundational theory for isolated hypersurface singularities over Discrete Valuation Rings. By treating the uniformizer as a variable and leveraging $p$ -derivations and specific uniformizer derivatives, the author generalizes classical results (Determinacy, Mather-Yau, Morse's Lemma) and provides precise numerical tools to detect and quantify singularities in mixed characteristic settings. This is a significant step forward for arithmetic geometry and the study of singularities in number-theoretic contexts.