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arxiv: 2604.26659 · v2 · submitted 2026-04-29 · 🧮 math.AG

Existence and maximal corank of simple Z_p-invariant germs

Ivan Proskurnin This is my paper

Pith reviewed 2026-05-08 03:18 UTC · model grok-4.3

classification 🧮 math.AG
keywords Z_p-invariant germscorankequivariant singularitiessimple germssingularity theoryprime order groupsasymptotic boundsalgebraic geometry
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The pith

The maximal corank of simple Z_p-invariant germs grows logarithmically to infinity as p increases, improving prior bounds on equivariantly stable singularities.

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

This paper improves the known upper bound on the corank of equivariantly stable singularities when the acting group has prime order. It further proves that the largest corank attained by any simple Z_p-invariant germ diverges to infinity with p and does so at a logarithmic rate, which shows that earlier bounds are asymptotically tight up to order of magnitude. A reader cares because corank controls the local complexity of singularities that appear in geometric problems with symmetry, so sharper control on their possible size directly affects classification results in algebraic geometry. The work therefore tightens the picture of what kinds of singularities can occur under cyclic prime-order symmetries.

Core claim

In this paper we improve the previously achieved upper bound on the corank of an equivariantly stable singularity for a group of prime order. We also prove that the maximal corank of a simple Z_p-invariant germ tends to infinity as p increases and is asymptotically logarithmic, so the previously obtained bound is valid up to order of magnitude.

What carries the argument

The corank of a simple Z_p-invariant germ, which quantifies the dimension of the space of linear terms in the equivariant normal form and is used to bound the complexity of singularities stable under prime-order cyclic group action.

If this is right

  • The upper bound on corank for equivariantly stable singularities under prime-order groups is tightened.
  • Simple Z_p-invariant germs exist with arbitrarily large corank once p is large enough.
  • The growth of this maximal corank is logarithmic in p, confirming that earlier estimates match the true order of magnitude.
  • The classification of such germs must accommodate singularities of unbounded complexity as the symmetry order increases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The logarithmic growth suggests that the number of distinct simple types also proliferates with p, though the paper does not count them.
  • Analogous logarithmic bounds may hold for other finite groups acting linearly, extending the technique beyond cyclic prime-order cases.
  • Explicit normal-form computations for moderately large p could provide concrete examples that saturate or approach the new bound.

Load-bearing premise

The standard notions of simplicity and equivariant stability for Z_p-invariant germs, together with the usual deformation theory in singularity classification, are taken to apply without exception for every prime p.

What would settle it

An explicit simple Z_p-invariant germ whose corank exceeds the improved upper bound for some prime p, or a family of such germs whose coranks grow faster than any multiple of log p, would falsify the claims.

read the original abstract

In this paper we improve the previously achieved upper bound on the corank of an equivariantly stable singularity for a group of prime order. We also prove that the maximal corank of a simple $\mathbb{Z}_p$-invariant germ tends to infinity as $p$ increases and is asymptotically logarithmic, so the previously obtained bound is valid up to order of magnitude.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The paper claims to improve the previously known upper bound on the corank of equivariantly stable singularities for groups of prime order Z_p. It further proves that the maximal corank of simple Z_p-invariant germs tends to infinity as p increases and grows asymptotically like log p, showing that the earlier bound is sharp up to order of magnitude.

Significance. If the proofs hold, the work advances equivariant singularity theory by sharpening corank estimates for Z_p-actions and establishing a logarithmic growth law for the maximal corank of simple invariants. The existence result combined with the asymptotic statement provides both an improved quantitative bound and confirmation of its near-optimality, which may facilitate further classification results in the area.

minor comments (1)
  1. Abstract: the specific numerical improvement to the previous upper bound is not stated explicitly; adding the new bound (or at least the factor by which it improves the old one) would allow readers to gauge the advance immediately.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of our results on improving the upper bound for the corank of equivariantly stable singularities under Z_p-actions, and the recommendation for minor revision. We appreciate the recognition that our existence result combined with the logarithmic asymptotic provides both a sharpened quantitative bound and confirmation of near-optimality.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives improved upper bounds on corank for equivariantly stable Z_p-singularities and proves logarithmic asymptotic growth of maximal corank for simple Z_p-invariant germs. These results follow from standard constructions in equivariant singularity theory (group actions on jet spaces, finite determinacy, orbit classification under Z_p), without any reduction of claims to fitted parameters, self-definitions, or load-bearing self-citations. The logical chain from group representation to corank estimates is independent and externally grounded in prior mathematical literature.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review based solely on abstract; no explicit free parameters, axioms, or invented entities are identifiable. The work necessarily rests on background results from equivariant singularity theory whose precise statements are unavailable.

pith-pipeline@v0.9.0 · 5340 in / 1036 out tokens · 26774 ms · 2026-05-08T03:18:00.302011+00:00 · methodology

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Reference graph

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