Relative Mather discrepancy on arc spaces

Tommaso de Fernex; Zach Mere

REVIEW 3 major objections 4 minor 1 cited by

Relative Mather discrepancy on arc spaces

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that a relative Mather discrepancy function on arc spaces computes the dimension of the kernel of the differential of the induced arc-space map, and that $\widehat K$-equivalent varieties share a motivic class.

desk verdict A plausible and promising paper on relative Mather discrepancy and \hat K-equivalence, but the supplied text is corrupted, so the proof claims are unverified rather than confirmed. read the letter →

arxiv 2508.12420 v4 pith:6L5AC3FX submitted 2025-08-17 math.AG

Tommaso de Fernex , Zach Mere This is my paper · ORCID

classification math.AG MSC 14E1814B05

keywords relativeMatherdiscrepancyarcspacesmotivicintegrationchange-of-variableformulaK-equivalenceringgenericallyétalemorphismssingularities

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to make the relative Mather discrepancy a usable invariant on the arc space of a singular variety. For any generically étale morphism $f:X\to Y$, it defines a discrepancy function on $X_\infty$ and proves that its value at an arc equals the dimension of the kernel of the differential of the induced map $f_\infty:X_\infty\to Y_\infty$. The paper also introduces a characteristic-free version of $K$-equivalence, called $\widehat K$-equivalence, and proves that $\widehat K$-equivalent varieties define the same class in the motivic ring. A reader should care because this gives a concrete geometric reading of arc-space fibers and extends birational equivalence invariants to positive characteristic.

What carries the argument

The central objects are the arc space $X_\infty$, the induced morphism $f_\infty:X_\infty\to Y_\infty$, and the relative Mather discrepancy function defined on $X_\infty$: it records, at each arc, the discrepancy between the relative Jacobian and the Mather class, and the paper's identity is that its value equals the dimension of the kernel of the differential $df_\infty$ at that arc. The change-of-variable formula is the bridge that turns this local identity into a global equality in the motivic ring.

What would settle it

Take a concrete singular generically étale map, such as the normalization of a cusp or a node, compute the relative Mather discrepancy at a family of arcs covering the singular point, and compare with the dimensions of the kernels of $df_\infty$ at those arcs; any arc where the two numbers differ would disprove the main theorem. Similarly, find two $\widehat K$-equivalent varieties whose motivic classes can be shown to differ by an explicit invariant.

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Extended reading notes

Core claim

The central discovery is that relative Mather discrepancy is not just a measure of how singular a variety is along an arc; it carries exact infinitesimal information about the arc-space map. Given a generically étale morphism $f:X\to Y$, the relative Mather discrepancy function on $X_\infty$ is shown to compute $\dim \ker df_\infty$ at every arc. The same discrepancy enters the change-of-variable formula of motivic integration, which is why the paper can pass from this pointwise computation to the global statement that $\widehat K$-equivalent varieties have equal motivic classes, for arbitrary characteristic.

Load-bearing premise

The definition of the relative Mather discrepancy must be well-behaved for every generically étale morphism of varieties in arbitrary characteristic, which requires finiteness of the relevant discrepancy loci and a working change-of-variable formula; if those conditions fail, the equality with the kernel dimension may fail.

Editorial extensions

If this is right

The kernel-dimension formula makes the relative Mather discrepancy a direct measure of the fibers of the arc-space map, so it can be read off from arc differentials rather than from a resolution.
Via the change-of-variable formula, the result ties the pointwise discrepancy to motivic integrals, giving a path from local arc data to global birational invariants.
Because $\widehat K$-equivalence works in arbitrary characteristic, the motivic-class equality holds for singular varieties over imperfect fields, where classical $K$-equivalence statements often require smoothness or resolution.
The definition supplies a new invariant that can distinguish varieties that are not $\widehat K$-equivalent.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial: if the same discrepancy function can be defined for non-generically étale morphisms, the kernel-dimension interpretation would likely need fiber-by-fiber corrections; a natural test is to factor an arbitrary dominant morphism into a generically étale part and a purely inseparable part and compare the two discrepancies.
Editorial: one may test whether $\widehat K$-equivalence classes are stable under products with affine space, a property expected of birational motivic invariants.
Editorial: the pointwise identity suggests an arc-level version of a dimension-defect formula, where the relative Mather discrepancy plays the role of a local rank defect; searching for such a formula for families of arcs would be a concrete next step.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a real paper by people who know this terrain, but I could not read the body. The supplied full text is mojibake. So what follows is judgment on the abstract plus scattered fragments, not on the proofs.

What's new: the relative Mather discrepancy function on X\infty for generically \'etale f: X\to Y, and the theorem that it computes the dimension of the kernel of df_\infty, is a genuinely new-looking arc-space invariant. The \hat K-equivalence notion is also new, and the fact that it matches K-equivalence on smooth varieties and gives equality in the motivic ring in arbitrary characteristic is a useful extension. Connecting this to the change-of-variable formula is the right framing.

What the paper does well, as far as I can see: the abstract is clear, the claims are concrete, and there is no visible circularity. The authors are not over-reaching in the abstract; they state exactly what the function computes and what the equivalence does.

Soft spots: all of them are about verification. The abstract omits the technical hypotheses needed: what precisely "generically \'etale" is doing, what conditions on singularity are needed for the discrepancy loci to be finite, and whether the change-of-variable formula has been established in arbitrary characteristic for the relevant class of morphisms. The arc space X_\infty is infinite-dimensional, so "dimension of the kernel of the differential" must be defined with care; presumably it is a constructible function or a codimension, but that is not in the abstract. None of these are named flaws; they are exactly the places a referee should push.

The stress-test note is accurate: no concrete error can be named, but no proof can be checked. The reader's UNVERDICTED, low-confidence verdict is the right epistemic state.

Who this is for: people working in arc spaces, motivic integration, and birational geometry, especially those interested in Mather discrepancy and K-equivalence extensions. If the proofs hold, the kernel-dimension theorem will be citable.

Recommendation: yes, send this to peer review. Ask the authors for a clean PDF and assign a referee who knows motivic integration. I would not desk reject it; the payoff justifies referee time.

Referee Report

3 major / 4 minor

Summary. The manuscript, as represented by its abstract, defines a relative Mather discrepancy function on the arc space X_∞ for any generically étale morphism f: X → Y, and claims that this function computes the dimension of the kernel of the differential of the induced morphism f_∞: X_∞ → Y_∞. It further introduces the notion of \hat K-equivalence, which agrees with K-equivalence for smooth varieties, and asserts that \hat K-equivalent varieties of arbitrary characteristic define the same class in the motivic ring. The supplied full text is corrupted beyond useable resolution: the body consists of unreadable encoded characters and fragmentary formulas, so no definitions, theorem statements, or proofs beyond the abstract can be recovered or verified.

Significance. If the main theorem is correct, it would supply a concrete geometric invariant on arc spaces and a characteristic-free extension of K-equivalence, with potential applications to motivic integration and singularity theory. The claimed equality between a discrepancy function and the kernel dimension of the differential of an arc-space map is natural and potentially useful. However, because the full text as provided is unreadable, the correctness of these claims is entirely unverified; the significance is therefore conditional on the missing proofs being valid.

major comments (3)

[Full text (as supplied)] The body of the manuscript as supplied is unreadable: it consists of mojibake and fragmented formulas, with no recoverable definitions, theorem statements, or proof text. The central claims of the abstract, namely the kernel-dimension formula for the relative Mather discrepancy and the statement about \hat K-equivalence, are therefore not verifiable from this version, so I cannot assess the mathematical soundness of the paper.
[Abstract] The abstract asserts that a relative Mather discrepancy function is defined on X_∞ for an arbitrary generically étale morphism, but it does not state the technical hypotheses that would make this well defined: for example, finiteness of the relevant discrepancy loci, a working change-of-variable formula in arbitrary characteristic, and the integrability conditions needed for the motivic-ring statement. These hypotheses are load-bearing and must be stated and proved; the unreadable body does not allow me to check whether they are satisfied.
[Full text (as supplied)] The supplied text contains a header referencing arXiv:2508.12419v1 [q-fin.PR], which does not match the paper's stated identifier (arXiv:2508.12420, math.AG). This mismatch is additional evidence that the provided file is corrupted rather than the intended submission, and it must be resolved before any mathematical evaluation can take place.

minor comments (4)

[Formatting] Please provide a clean, readable copy of the manuscript; the current version is unusable.
[Abstract] The promised relation to the change-of-variable formula in motivic integration is not visible in the supplied text; a precise statement with equation numbers would help readers locate it in the body.
[Definitions] The notation \hat K-equivalence is introduced in the abstract, but no formal definition is recoverable; the body should include the definition and a comparison with the existing K-equivalence for smooth varieties.
[Examples] No concrete examples or applications are readable; a few explicit computations of the relative Mather discrepancy for simple singular varieties would illustrate the definition and make the paper more accessible.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity can be identified from the recoverable text: the body is corrupted, and the abstract alone exhibits no reduction of a claimed result to its own inputs.

full rationale

The only legible parts of the manuscript are the abstract, some section headings, and fragments of displayed formulas; the body text is garbled to the point that definitions, proofs, and citations cannot be inspected. The abstract states that the relative Mather discrepancy function is defined and shown to compute the kernel dimension of the differential of the induced arc-space morphism, and that a new K-hat-equivalence relation implies equality of motivic classes. Since the defining equations, the statement of the change-of-variable formula, and the proof chain are not recoverable, I cannot exhibit any specific step in which a claimed prediction is equivalent by construction to a fitted input or to a self-citation. Absent such an exhibited reduction, the appropriate finding is no circularity, not a speculative accusation. The paper is unverified in this rendering, but unverified is not the same as circular.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Only the abstract is available and it states no explicit free parameters or axioms. The single domain assumption is the generically étale hypothesis. No invented entities are introduced; the new definitions are mathematical constructions, not postulated entities.

assumptions (1)

domain assumption The morphism f: X -> Y is generically étale.
Stated in the abstract as the setting for the main theorem; the relative Mather discrepancy function and the kernel-dimension computation are defined only for such morphisms.

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Cite this review

Pith. "Pith review of Relative Mather discrepancy on arc spaces." pith.science (2026). https://pith.science/paper/6L5AC3FX

@misc{pith2026250812420,
  author       = {Pith},
  title        = {Pith review of: Relative Mather discrepancy on arc spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6L5AC3FX}},
  note         = {Machine review of arXiv:2508.12420}
}

abstract

Given any generically \'etale morphism of varieties $f \colon X \to Y$, we define the relative Mather discrepancy function on the arc space $X_\infty$ of the domain and show that this function computes the dimension of the kernel of the differential map of the induced morphism on arc spaces $f_\infty \colon X_\infty \to Y_\infty$. We relate this result to the change-of-variable formula in motivic integration. We introduce the notion of $\widehat K$-equivalence, which agrees with $K$-equivalence for smooth varieties, and prove that $\widehat K$-equivalent varieties of arbitrary characteristic define the same class in the motivic ring.

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