pith. sign in

arxiv: 1906.11545 · v1 · pith:UGK3DNXFnew · submitted 2019-06-27 · 🧮 math.DG

A step to Gronwall's conjecture

Jean Paul Dufour This is my paper

Pith reviewed 2026-05-25 14:41 UTC · model grok-4.3

classification 🧮 math.DG
keywords plane linear 3-websGronwall conjecturecharacteristic invariantlocal homographydiffeomorphismnon-zero curvaturegeneric pointsweb geometry
0
0 comments X

The pith

A diffeomorphism between linear 3-webs that preserves the characteristic at a point is locally a homography.

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

The paper defines an invariant called the characteristic attached to each generic point of a plane linear 3-web with non-zero curvature. It establishes that any diffeomorphism mapping one such web to another while sending a point to a point with matching characteristic value must be a local homography. This supplies a concrete step toward Gronwall's conjecture, which asserts that every local isomorphism between two such webs is necessarily a homography. A sympathetic reader would care because the result isolates an invariant whose preservation under a map forces the map to be projective, thereby narrowing the possible local equivalences between these geometric structures.

Core claim

Using recent results on plane linear 3-webs, the paper attaches to each generic point an invariant called the characteristic. The central property shown is that a diffeomorphism interchanging two linear 3-webs with non-zero curvature and sending a point of the first to a point of the second with the same characteristic is locally a homography.

What carries the argument

The characteristic, an invariant attached to generic points of a linear 3-web with non-zero curvature, which is preserved by homographies and whose preservation by a diffeomorphism forces that diffeomorphism to be a local homography.

If this is right

  • Any diffeomorphism between two linear 3-webs that preserves the characteristic at one corresponding point is necessarily a local homography.
  • The characteristic supplies a sufficient condition for an isomorphism between such webs to be projective.
  • The result applies at generic points of webs possessing non-zero curvature.
  • The approach isolates the role of this single invariant in forcing the homography conclusion.

Where Pith is reading between the lines

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

  • If every local isomorphism between linear 3-webs could be shown to preserve the characteristic, the full Gronwall conjecture would follow at once.
  • The invariant may be used to distinguish pairs of webs that cannot be related by any homography.
  • Analogous invariants might be constructed for other classes of webs to classify their local automorphism groups.

Load-bearing premise

The construction and verification of the characteristic rest on the validity of prior results about plane linear 3-webs together with the assumption that the points are generic and have non-zero curvature.

What would settle it

An explicit pair of plane linear 3-webs with non-zero curvature together with a diffeomorphism mapping a generic point to another point of equal characteristic yet failing to be a local homography would disprove the stated property.

read the original abstract

In this paper we will explore a way to prove the hundred years old Gronwall's conjecture: if two plane linear 3-webs with non-zero curvature are locally isomorphic, then the isomorphism is a homography. Using recent results of S. I. Agafonov, we exhibit an invariant, the {\sl characteristic}, attached to each generic point of such a web, with the following property: if a diffeomorphism interchanges two such linear webs, sending a point of the first to a point of the second which have the same characteristic, then this diffeomomorphism is locally a homography.

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 proposes an approach to Gronwall's conjecture on plane linear 3-webs: if two such webs with non-zero curvature are locally isomorphic, the isomorphism is a homography. Invoking theorems of Agafonov on linear 3-webs, the paper constructs an invariant called the characteristic at generic points and establishes that any diffeomorphism interchanging two webs while sending a point to one with matching characteristic must be a local homography.

Significance. If the construction and invariance of the characteristic hold, the result supplies a concrete, pointwise invariant that forces candidate isomorphisms to be homographies, constituting a useful incremental step toward the conjecture. The approach is parameter-free in the sense that it attaches the invariant directly via the cited web geometry, but its overall significance is reduced by the absence of independent verification of the Agafonov input.

major comments (2)
  1. [Introduction] The construction of the characteristic (Introduction and the paragraph following the statement of the main property) is defined by direct appeal to Agafonov's theorems on plane linear 3-webs without restating the relevant statements, definitions of the web curvature, or the steps that establish invariance under the diffeomorphism. Consequently the central claim that matching characteristics imply the map is a local homography inherits any scope restrictions or unstated genericity assumptions present in the cited results.
  2. The manuscript states that the points considered are generic with non-zero curvature but supplies no explicit verification that the characteristic remains well-defined or that the homography conclusion continues to hold when the curvature vanishes or at non-generic points; this leaves open whether the exhibited property covers the full statement of Gronwall's conjecture.
minor comments (2)
  1. Notation for the characteristic is introduced without an equation label or explicit formula; adding a displayed definition would improve readability.
  2. The abstract and introduction both refer to 'recent results of S. I. Agafonov' but the reference list entry should include the precise title, journal, and year for the cited work.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's detailed feedback on our manuscript. Below we respond point by point to the major comments, indicating planned revisions where appropriate. Our goal is to strengthen the presentation while maintaining the paper's focus as an incremental step toward Gronwall's conjecture.

read point-by-point responses

  1. Referee: [Introduction] The construction of the characteristic (Introduction and the paragraph following the statement of the main property) is defined by direct appeal to Agafonov's theorems on plane linear 3-webs without restating the relevant statements, definitions of the web curvature, or the steps that establish invariance under the diffeomorphism. Consequently the central claim that matching characteristics imply the map is a local homography inherits any scope restrictions or unstated genericity assumptions present in the cited results.

    Authors: We agree that the manuscript would be improved by greater self-containment. In the revised version, we will include a brief summary of the relevant definitions from web geometry, including the web curvature, and restate the key theorems of Agafonov that are invoked. We will also outline the steps establishing that the characteristic is invariant under diffeomorphisms preserving the web structure. This will make the scope and assumptions explicit without altering the core argument. revision: yes

  2. Referee: [—] The manuscript states that the points considered are generic with non-zero curvature but supplies no explicit verification that the characteristic remains well-defined or that the homography conclusion continues to hold when the curvature vanishes or at non-generic points; this leaves open whether the exhibited property covers the full statement of Gronwall's conjecture.

    Authors: The statement of Gronwall's conjecture in the manuscript applies specifically to linear 3-webs with non-zero curvature, and our result is formulated for generic points where the characteristic is defined via Agafonov's theorems. We do not assert that the characteristic is well-defined or that the conclusion holds when curvature vanishes or at non-generic points, as these cases fall outside the conjecture's hypotheses or require separate treatment. To clarify this, we will add a remark in the introduction specifying the scope of the result. However, we maintain that the paper provides a useful invariant for the generic case, which is a step toward the full conjecture. revision: partial

Circularity Check

0 steps flagged

No circularity; central step relies on external Agafonov theorems

full rationale

The paper states its approach explicitly: 'Using recent results of S. I. Agafonov, we exhibit an invariant, the characteristic...' (abstract). The derivation chain begins with the external theorems on plane linear 3-webs to attach the characteristic at generic non-zero-curvature points, then deduces the homography property for diffeomorphisms preserving it. No self-citation occurs (Agafonov is a distinct author), no parameter is fitted then renamed as prediction, no ansatz is smuggled via prior work by the same authors, and no uniqueness theorem is imported from the present author's earlier papers. The result therefore does not reduce to its own inputs by construction; it is a standard dependence on cited external results. This matches the default expectation of no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence of the characteristic invariant whose definition and key property are taken from Agafonov's prior results on 3-webs; no free parameters or new entities beyond the invariant itself are mentioned.

axioms (1)
  • domain assumption Recent results of S. I. Agafonov on plane linear 3-webs with non-zero curvature
    The paper states it uses these results to attach the characteristic to generic points.
invented entities (1)
  • characteristic no independent evidence
    purpose: Invariant attached to each generic point of a linear 3-web to force diffeomorphisms to be homographies when matched
    Newly introduced in the paper as the tool for the stated property.

pith-pipeline@v0.9.0 · 5611 in / 1234 out tokens · 28825 ms · 2026-05-25T14:41:44.584694+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

6 extracted references · 6 canonical work pages · 1 internal anchor

  1. [1]

    Projective invariants of linear 3-webs and Gronwall's Conjecture

    S.I. Agafonov, Projective invariants of linear 3-web and Gronwall's Conjecture. arXiv 1708.01996v2 [mathsDG], 2019

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  2. [2]

    Blaschke, Uber die Tangenten einer ebenen Kurve funfter Klasse

    W. Blaschke, Uber die Tangenten einer ebenen Kurve funfter Klasse. Abh. Math. Semin. Hamb. Univ. 9 (1933) 313-317

    work page 1933

  3. [3]

    J. P. Dufour, P. Jean Rigidity of Webs and Families of Hypersurfaces. in Singularities and Dynamical Systems, (Iraklion, 1983), North Holland, Amsterdam, (1985), 271-283

    work page 1983

  4. [4]

    T. H. Gronwall, Sur les \'equations entre trois variables repr\'esentables par les nomogrammes \`a points align\'es. Journal de Liouville, 8, (1912),59-102

    work page 1912

  5. [5]

    H\'enaut, Analytic web geometry , 6-48, in Web theory and related topics

    A. H\'enaut, Analytic web geometry , 6-48, in Web theory and related topics. (J. Grifone and E. Salem editors) World Scientific Publishing Co. Ptc. Ltd 2001

    work page 2001

  6. [6]

    Sternberg, On the structure of local homeomorphisms of euclidean n space

    S. Sternberg, On the structure of local homeomorphisms of euclidean n space. Ann. of Math. 80, (1958), 623-631

    work page 1958