pith. sign in

arxiv: 2604.14729 · v2 · submitted 2026-04-16 · 🧮 math.AG

A remark on isolated complex hypersurface singularities

Fabrizio Catanese , Ciro Ciliberto , Concettina Galati This is my paper

Pith reviewed 2026-05-10 09:56 UTC · model grok-4.3

classification 🧮 math.AG
keywords hypersurface singularitiesanalytic equivalencetangent coneordinary multiple pointregular polynomialquasihomogeneous polynomialsisolated singularities
0
0 comments X

The pith

For regular homogeneous polynomials of degree m, perturbations of order at least n(m-2)+1 yield germs analytically equivalent to the tangent cone.

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

The paper addresses whether a hypersurface germ with an ordinary multiple point at the origin is analytically the same as its tangent cone. While this fails in general, the note proves that when the tangent cone is defined by a regular homogeneous polynomial (corresponding to a smooth projective hypersurface), there exists an explicit integer threshold D(n,m) equal to n(m-2) plus one. Any convergent series g that agrees with f up to order d-1 where d meets or exceeds this threshold produces a germ equivalent to that of f. This provides a concrete way to recognize when the analytic structure is fixed by the lowest degree terms alone, with an extension noted for quasihomogeneous cases.

Core claim

There is an integer D(n,m) = n(m-2) + 1 such that for a regular homogeneous polynomial f of degree m, any g = f + o(d) with d >= D(n,m) defines a germ analytically equivalent to {f=0}.

What carries the argument

The threshold D(n,m) = n(m-2)+1, serving as the minimal order guaranteeing that higher terms cannot change the analytic isomorphism class when the initial form is regular.

If this is right

  • If the order of perturbation is high enough, the analytic type of the singularity is completely determined by its tangent cone for regular cones.
  • The explicit formula allows verification by checking only finitely many coefficients in the power series expansion.
  • The result extends to quasihomogeneous polynomials, broadening the class of singularities where the tangent cone determines the germ.
  • Since the bound was previously known as an exercise, this note provides an expository derivation and clarification.

Where Pith is reading between the lines

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

  • This criterion could be used to simplify computations in the classification of isolated hypersurface singularities by reducing the problem to checking the tangent cone after a certain order.
  • One could test the sharpness of the bound by constructing examples where a lower-order perturbation changes the analytic type for a regular f.
  • Connections to deformation theory suggest that this stability implies the singularity is rigid in certain directions of the versal deformation space.
  • The approach might generalize to other types of singularities beyond hypersurfaces.

Load-bearing premise

The projective hypersurface defined by the homogeneous polynomial f must be smooth.

What would settle it

A counterexample consisting of a smooth projective hypersurface of degree m in P^{n-1} and a power series g agreeing with f to order less than n(m-2) but where the germ of g=0 is not analytically equivalent to f=0.

read the original abstract

This is now an expository note about the following classical problem. Let $(X, \bf 0)$ be the germ of a hypersurface in $(\mathbb C^n,\bf 0)$ with an ordinary singularity of multiplicity $m$ at the origin $\bf 0$. A natural question to ask is whether $X$ and its tangent cone at the origin are analytically isomorphic. The answer is negative in general, in view of a theorem of Kioji Saito. However there is an integer $D(n,m)>m$ such that, given a \emph{regular} homogeneous polynomial $f(x_1,\ldots, x_n)$ of degree $m$ (this means that $\{ f=0\}$ is a smooth hypersurface in $\PP^{n-1}$) then, for all $d\geq D(n,m)$, any convergent power series of the form $g=f+ o(d)$ (here, as usual, $o(d)$ stays for a power series of order at least $d$), defines a germ $\{ g=0\}$ which is analytically equivalent to the germ $\{ f=0\}$. In this note we compute $D(n,m)$ explicitly as $n(m-2)+1$. We also give an extension to the case in which $f$ is a quasihomogeneous polynomial. It was pointed out that the value of $D(n,m)$ was already known by \cite[Exercise 7.31]{D}.

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

2 major / 2 minor

Summary. The manuscript is an expository note on analytic equivalence of isolated hypersurface germs with ordinary multiplicity-m singularities. It recalls Saito's theorem showing that equivalence to the tangent cone fails in general, then asserts that when the tangent cone is given by a regular (smooth projective) homogeneous polynomial f of degree m, there exists an explicit integer D(n,m) = n(m-2)+1 > m such that any convergent g = f + o(d) with d ≥ D yields a germ analytically equivalent to {f=0}. An extension to quasihomogeneous f is sketched, and a citation to a prior exercise is noted.

Significance. A correct explicit bound D(n,m) would be a useful, computable refinement of the classical theory of ordinary singularities, giving a concrete threshold beyond which higher-order perturbations cannot change the analytic type when the tangent cone is smooth. The attempt to furnish an explicit formula and the quasihomogeneous extension are positive features; the note is short and focused.

major comments (2)
  1. [Abstract] Abstract (and the paragraph stating the main result): the claimed explicit value D(n,m) = n(m-2)+1 fails to satisfy the required inequality D(n,m) > m for all integers n ≥ 2, m ≥ 2. When m = 2 one obtains D = 1 < 2; when n = 2 and m = 3 one obtains D = 3 which is not strictly greater than 3. Because analytic equivalence preserves multiplicity, a perturbation of order d = 1 allows g to have multiplicity 1 while {f=0} has multiplicity 2, so equivalence is impossible. This directly contradicts the central claim that the stated D works for all d ≥ D.
  2. [Main result paragraph] The derivation or citation of the formula (referenced to Exercise 7.31 in [D]): the manuscript must either correct the expression for D(n,m) (e.g., by taking the maximum with m+1 or adding hypotheses on n and m) or prove that the given expression satisfies D > m under the standing assumptions. The current mismatch is load-bearing for the explicit-computation assertion.
minor comments (2)
  1. [Abstract] The abbreviation 'o(d)' is used without an explicit reminder that it denotes a power series of order at least d; a parenthetical clarification would help readers outside singularity theory.
  2. The reference [D] should appear with full bibliographic details in the bibliography.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the inconsistency between the claimed inequality D(n,m) > m and the explicit formula n(m-2)+1. We agree that the stated bound fails to satisfy the inequality in some cases (such as m=2 or n=2, m=3), which is necessary to preserve multiplicity under analytic equivalence. We will revise the manuscript to define D(n,m) as max(n(m-2)+1, m+1), thereby correcting the central claim while retaining the explicit and computable character of the result.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and the paragraph stating the main result): the claimed explicit value D(n,m) = n(m-2)+1 fails to satisfy the required inequality D(n,m) > m for all integers n ≥ 2, m ≥ 2. When m = 2 one obtains D = 1 < 2; when n = 2 and m = 3 one obtains D = 3 which is not strictly greater than 3. Because analytic equivalence preserves multiplicity, a perturbation of order d = 1 allows g to have multiplicity 1 while {f=0} has multiplicity 2, so equivalence is impossible. This directly contradicts the central claim that the stated D works for all d ≥ D.

    Authors: We agree with this assessment. The formula n(m-2)+1 does not guarantee D > m for all admissible n and m, and the multiplicity-preservation argument is correct and fundamental. We will revise the abstract to state that D(n,m) equals the maximum of n(m-2)+1 and m+1. This ensures the inequality holds while keeping the bound explicit and directly tied to the classical theory of ordinary singularities. revision: yes

  2. Referee: [Main result paragraph] The derivation or citation of the formula (referenced to Exercise 7.31 in [D]): the manuscript must either correct the expression for D(n,m) (e.g., by taking the maximum with m+1 or adding hypotheses on n and m) or prove that the given expression satisfies D > m under the standing assumptions. The current mismatch is load-bearing for the explicit-computation assertion.

    Authors: We accept the referee's recommendation and will correct the expression rather than add restrictive hypotheses. The revised manuscript will define D(n,m) := max(n(m-2)+1, m+1) and will clarify that Exercise 7.31 in [D] supplies the base bound n(m-2)+1, which we adjust by the multiplicity requirement m+1. The main result paragraph and the surrounding discussion will be updated accordingly. revision: yes

Circularity Check

0 steps flagged

No circularity; bound derived from external classical reference

full rationale

The paper is an expository note recalling a classical result on analytic equivalence of hypersurface germs with ordinary singularities. It states the existence of an integer D(n,m)>m with the stated property for regular homogeneous f of degree m, then asserts that this D equals n(m-2)+1, noting that the value was already known from an external source (Exercise 7.31 in [D]). No derivation step reduces the claimed bound to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation; the argument rests on prior literature in singularity theory without internal circular reduction. The noted mismatch between the formula and the strict inequality D>m for small m (e.g., m=2) is a potential error in the stated range of applicability, not evidence that any step equates output to input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard facts from complex analytic geometry (convergent power series, analytic equivalence of germs) and the definition of regularity for homogeneous polynomials; no free parameters or invented entities are introduced.

axioms (2)
  • standard math Convergent power series rings over C admit well-defined notions of order and analytic isomorphism of germs.
    Invoked throughout the statement of the equivalence for g = f + o(d).
  • domain assumption A homogeneous polynomial f is regular when the projective hypersurface it defines in PP^{n-1} is smooth.
    Used to guarantee the analytic equivalence once the perturbation order meets or exceeds D(n,m).

pith-pipeline@v0.9.0 · 5573 in / 1436 out tokens · 50742 ms · 2026-05-10T09:56:51.118942+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

  1. [1]

    I.; Gusein-Zade, S

    Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N.Singularities of differentiable maps. Vol. I“Nauka”, Moscow, 1982, 304 pp., republished in Monogr. Math., 82 Birkh¨ auser Boston, Inc., Boston, MA, 1985, xi+382 pp

    work page 1982

  2. [2]

    Catanese,Complex Manifolds, G¨ ottingen (1999), preliminary version, 1-144 pages, Lecture notes taken by P

    F. Catanese,Complex Manifolds, G¨ ottingen (1999), preliminary version, 1-144 pages, Lecture notes taken by P. Frediani

    work page 1999

  3. [3]

    Ciliberto, C

    C. Ciliberto, C. Galati,On nodal deformations of singular surfaces inP 3, arXiv:2602.09177, (2026)

    work page arXiv 2026

  4. [4]

    Dimca,Topics on real and complex singularities

    A. Dimca,Topics on real and complex singularities. An introduction.Adv. Lectures Math. Friedr. Vieweg & Sohn, Braunschweig, 1987, xviii+242 pp

    work page 1987

  5. [5]

    Franchetta,Osservazioni sui punti doppi isolati delle superficie algebriche, Rend

    A. Franchetta,Osservazioni sui punti doppi isolati delle superficie algebriche, Rend. di Mat. e Appl.,5, (1946), 283–290. A REMARK ON ISOLATED COMPLEX HYPERSURFACE SINGULARITIES 7

    work page 1946

  6. [6]

    Greuel, C

    G.-M. Greuel, C. Lossen, E. Shustin:Introduction to Singularities and Deformations, First edition, Springer, Berlin, 2007, xii+471 pp. Second edition, Springer Monographs in Mathematics, 2025, xvii+717 pp

    work page 2007

  7. [7]

    Saito,Quasihomogene isolierte Singularit¨ aten von Hyperfl¨ achen, Invent

    K. Saito,Quasihomogene isolierte Singularit¨ aten von Hyperfl¨ achen, Invent. Math.,14, (1971), 123–142

    work page 1971

  8. [8]

    Scheja, U

    G. Scheja, U. Storch,Uber Spurfunktionen bei vollst¨ andigen Durchschnitten, J. Reine Angew. Math.278/279(1975), 174–189

    work page 1975

  9. [9]

    van Straten, T

    D. van Straten, T. Warmt,Gorenstein-duality for one-dimensional almost complete intersec- tions with an application to non-isolated real singularities, Math. Proc. Cambridge Philos. Soc., 158, (2015), no. 2, 249–268

    work page 2015

  10. [10]

    Tor Vergata

    J. C. Tougeron,Id´eaux de fonctions diff´erentiables, Ann. Inst. Fourier,18, (1968), 177–240. Mathematisches Institut, Fakult¨at f¨ur Mathematik, Physik und Informatik, Univer- sit¨at Bayreuth Email address:fabrizio.catanese@uni-bayreuth.de Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy. Ema...

    work page 1968