The McKay correspondence and local heights for wild-by-tame split metacyclic groups
Pith reviewed 2026-05-08 16:34 UTC · model grok-4.3
The pith
For wild-by-tame split metacyclic groups, crepant resolutions of quotients have Euler characteristics depending on the representation chosen.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We calculate the stringy motive of the quotient variety and find a formula for its stringy Euler number. As a consequence, we prove that a crepant resolution of the quotient variety does not in general have Euler characteristic equal to the number of conjugacy classes in G, in contrast to the classical case. In particular, we show it depends on the choice of representation as well as the group. As part of this, we compute the v-function associated to a G-representation, corresponding to a stacky local height function.
What carries the argument
The stringy motive of the quotient variety, computed via the v-function for the G-representation that functions as a stacky local height and yields the stringy Euler number.
Load-bearing premise
The stringy motive of the quotient can be computed explicitly for these wild-by-tame split metacyclic groups and that the resulting Euler number formula correctly captures the dependence on the representation.
What would settle it
For a specific wild-by-tame split metacyclic group and representation, construct a crepant resolution if it exists and compute its ordinary Euler characteristic to test whether it equals the stringy Euler number given by the formula.
read the original abstract
We study the McKay correspondence for the representations of certain wild-by-tame split metacyclic groups whose order is divisible by the characteristic of the base field. We calculate the stringy motive of the quotient variety and find a formula for its stringy Euler number. As a consequence, we prove that a crepant resolution of the quotient variety (provided one exists) does not in general have Euler characteristic equal to the number of conjugacy classes in $G$, in contrast to the classical case. In particular, we show it depends on the choice of representation as well as the group. As part of this, we compute the v-function associated to a $G$-representation, corresponding to a stacky local height function.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the McKay correspondence for wild-by-tame split metacyclic groups whose order is divisible by the base field characteristic. It computes the stringy motive of the quotient variety arising from a linear G-representation on affine space, derives an explicit formula for the associated stringy Euler number via the v-function, and shows that this number depends on the choice of representation. As a consequence, a crepant resolution (when it exists) does not in general have Euler characteristic equal to the number of conjugacy classes of G, in contrast to the tame case.
Significance. If the explicit computations hold, the result provides a concrete extension of the McKay correspondence into the wild setting, exhibiting a representation-dependent failure of the classical Euler characteristic equality. The introduction of the v-function as a stacky local height function supplies a new computational tool for stringy invariants in positive characteristic. The direct derivation of the Euler-number formula for this class of groups is a strength that allows falsifiable checks against specific examples.
minor comments (2)
- The abstract states the main results but does not sketch the strategy used to compute the stringy motive or v-function; adding one sentence on the overall approach would improve accessibility without altering the technical content.
- Notation for the v-function and its relation to the stringy motive is introduced in the main text; a short table or diagram summarizing the key formulas (e.g., how the Euler number is extracted) would aid readers following the dependence on the representation.
Simulated Author's Rebuttal
We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report accurately reflects the manuscript's focus on extending the McKay correspondence to wild-by-tame split metacyclic groups and the role of the v-function in computing stringy invariants. No specific major comments were listed in the report.
Circularity Check
No significant circularity detected
full rationale
The manuscript derives its central result by direct, explicit computation of the stringy motive of the quotient variety and the associated v-function for wild-by-tame split metacyclic groups. The stringy Euler number is obtained from these calculations and then used to exhibit the dependence on the choice of representation, yielding the stated contrast with the classical McKay correspondence. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the derivation chain consists of independent algebraic computations that stand on their own without circular reduction to the target claim.
Axiom & Free-Parameter Ledger
Reference graph
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