Confocal families of plane algebraic curves

Boris Shapiro; Ragni Piene

arxiv: 2604.25293 · v2 · pith:26ZEGW6Ynew · submitted 2026-04-28 · 🧮 math.AG · math.CV

Confocal families of plane algebraic curves

Ragni Piene , Boris Shapiro This is my paper

Pith reviewed 2026-05-21 00:29 UTC · model grok-4.3

classification 🧮 math.AG math.CV

keywords plane algebraic curvesconfocal familiesfocifocal mapequiclassical familiesdeformation theoryalgebraic geometry

0 comments

The pith

Families of plane algebraic curves sharing the same foci can be described by a focal map on equiclassical families whose fibers are analyzed using deformation theory.

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

The paper examines collections of plane algebraic curves that all possess an identical set of foci. It re-expresses the confocality condition by defining a focal map on equiclassical families of such curves. The structure of the fibers of this map is then investigated with standard techniques from deformation theory. A reader would care because this gives an organized algebraic framework for studying classical confocal configurations beyond low-degree cases.

Core claim

We study families of plane algebraic curves sharing the same set of foci. We reformulate confocality via a focal map on equiclassical families and analyze its fibers using deformation theory.

What carries the argument

The focal map on equiclassical families of plane algebraic curves, which encodes shared foci and whose fibers are examined for geometric structure.

If this is right

Confocal families correspond precisely to the fibers of the focal map.
Deformation theory determines the local structure and dimension of each fiber.
The map organizes all such families sharing a fixed set of foci into a coherent algebraic space.
Classical confocal properties extend to higher-degree plane curves through this construction.

Where Pith is reading between the lines

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

The method may adapt to confocal hypersurfaces in higher-dimensional projective space.
It could link to separation-of-variables problems where confocal quadrics appear in classical mechanics.
Low-degree explicit computations might confirm the expected fiber dimensions for specific foci sets.

Load-bearing premise

Confocality admits a reformulation as a focal map on equiclassical families such that the fibers of the map can be analyzed using standard deformation theory.

What would settle it

An explicit pair of confocal plane curves of the same degree where the dimension or smoothness of their joint deformation space fails to match the fiber dimension predicted by the focal map.

read the original abstract

We study families of plane algebraic curves sharing the same set of foci. We reformulate confocality via a focal map on equiclassical families and analyze its fibers using deformation theory.

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.

Desk Editor's Note private letter to a colleague

This paper reformulates confocality of plane algebraic curves as fibers of a focal map on equiclassical families and analyzes them with deformation theory.

read the letter

The punchline is that this paper reformulates confocality for families of plane algebraic curves by introducing a focal map on equiclassical families and then analyzes the fibers of that map with deformation theory. The authors look at curves sharing foci and set up this map to capture the confocality condition. Using deformation theory on the fibers makes sense for understanding the local structure or dimensions of these families. This seems like a reasonable way to bring classical geometry into a more modern algebraic setting. What the paper does well is to provide a structured framework without adding unnecessary complexity. Deformation theory is the right tool here, and defining the map on equiclassical families avoids some of the messier aspects of working directly with arbitrary curve families. Credit to the authors for keeping the focus on the geometry. The soft spots are limited. Since the provided text is mostly the abstract, I cannot check the explicit construction of the focal map or any sample calculations. That makes it tough to judge if the reformulation is substantially new or mostly a repackaging of known deformation results. The fiber analysis might be straightforward, which is fine if it is presented clearly. This paper is for algebraic geometers working on moduli of curves or classical special families. Someone interested in extending confocal properties beyond conics would find it useful. It is not for a general audience. The work shows clear thinking on its own terms and engages honestly with the tools of the field. I would bring it to a reading group if the group has people into curve geometry. It deserves a serious referee. The direction is solid and the methods fit the problem. My recommendation is to send it for peer review. It looks like the kind of paper that can be improved with feedback on the details but is worth the time.

Referee Report

0 major / 4 minor

Summary. The paper studies families of plane algebraic curves that share the same set of foci. It reformulates the notion of confocality by introducing a focal map defined on equiclassical families of such curves and then analyzes the fibers of this map using standard techniques from deformation theory.

Significance. If the reformulation and fiber analysis are correct, the work provides a useful bridge between classical confocal geometry and modern algebraic geometry, potentially allowing deformation-theoretic tools to classify or parametrize confocal families beyond the classical conic case. The approach appears coherent and could serve as a foundation for further study of equiclassical conditions in the plane.

minor comments (4)

The abstract is very brief; consider expanding the introduction to include a short statement of the main theorem or a concrete low-degree example (e.g., confocal cubics) that illustrates the focal map before the general construction.
Clarify the precise definition of 'equiclassical family' early in the manuscript; the term is used in the abstract but its relation to the usual notions of equisingular or equiclassical deformations should be stated explicitly with a reference if it is standard.
In the section describing the focal map, ensure that the target space of the map is defined with the same level of detail as the domain; a diagram or commutative square relating the focal map to the incidence variety would improve readability.
Add a brief discussion of the base field (presumably algebraically closed of characteristic zero) and any flatness or smoothness assumptions needed for the deformation theory arguments to apply without additional hypotheses.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation of minor revision. We are pleased that the reformulation of confocality via the focal map on equiclassical families, together with the deformation-theoretic analysis of its fibers, is recognized as a potential bridge between classical confocal geometry and modern algebraic geometry.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's abstract and described approach consist of studying confocal families of plane algebraic curves by reformulating confocality as a focal map on equiclassical families and then applying standard deformation theory to analyze the fibers. No equations, definitions, or self-citations are visible that would reduce any claimed result to its own inputs by construction. The central steps rely on external, standard techniques from algebraic geometry rather than any fitted parameters, self-referential definitions, or load-bearing prior results from the same authors. The derivation chain is therefore self-contained and independent.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the work appears to rest on standard background from algebraic geometry and deformation theory without new postulates visible here.

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We reformulate confocality via a focal map on equiclassical families and analyze its fibers using deformation theory.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The differential of the focal map is induced by restriction to the isotropic line L

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Plane rectifiable curves: old and new
math.AG 2026-05 unverdicted novelty 6.0

New criteria for algebraically rectifiable plane curves are introduced by relating them to quadratic differentials, with a generalization to differentials of higher order.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages · cited by 1 Pith paper

[1]

Mary Ruth Andrews,Foci of algebraic curves, Cornell University, 1941

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Arnold Emch,On plane algebraic curves with a given system of foci, Bull. Amer. Math. Soc.25(1919), no. 4, 157–161

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Markus Hunziker, Andrei Mart´ ınez-Finkelshtein, Taylor Poe, and Brian Simanek, Poncelet–Darboux, Kippenhahn, and Szeg˝ o: interactions between projective geometry, matrices and orthogonal polynomials, J. Math. Anal. Appl.511(2022), no. 1, Paper No. 126049, 35

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George Salmon,A treatise on the higher plane curves: intended as a sequel to “A treatise on conic sections”, Chelsea Publishing Co., New York, 1960. 3rd ed

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Oscar Zariski,Algebraic surfaces, supplemented edition, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. Band 61, Springer-Verlag, New York-Heidelberg, 1971. With appendices by S. S. Abhyankar, J. Lipman, and D. Mumford. Ragni Piene, Department of Mathematics, University of Oslo, P.O.Box 1053 Blindern, NO-0...

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[1] [1]

Mary Ruth Andrews,Foci of algebraic curves, Cornell University, 1941

work page 1941

[2] [2]

Alfred Barnard Basset,An elementary treatise on cubic and quartic curves, Cam- bridge; Deighton, Bell, 1901

work page 1901

[3] [3]

Scand.111(2012), no

Eduardo Casas-Alvero,Siebeck curves and two refinements of the Gauss-Lucas theo- rem, Math. Scand.111(2012), no. 1, 12–41

work page 2012

[4] [4]

,Roots of complex polynomials and foci of real algebraic curves, Enseign. Math. (2)58(2012), no. 3-4, 223–248

work page 2012

[5] [5]

Volume 6, Cambridge Library Col- lection, Cambridge University Press, Cambridge, 2009

Arthur Cayley,The collected mathematical papers. Volume 6, Cambridge Library Col- lection, Cambridge University Press, Cambridge, 2009. Reprint of the 1893 original

work page 2009

[6] [6]

Julian Lowell Coolidge,A treatise on algebraic plane curves, Dover Publications, Inc., New York, 1959

work page 1959

[7] [7]

Arnold Emch,On plane algebraic curves with a given system of foci, Bull. Amer. Math. Soc.25(1919), no. 4, 157–161

work page 1919

[8] [8]

James Gow,A short history of Greek mathematics, Cambridge University Press, 1884

work page

[9] [9]

Harold Hilton,Plane algebraic curves, Oxford University Press, 1932

work page 1932

[10] [10]

Jervis,On the real foci and directrices of a class cubic and other plane algebraic curves, Proc

Harold Hilton and Sybil D. Jervis,On the real foci and directrices of a class cubic and other plane algebraic curves, Proc. London Math. Soc. (2)27(1928), no. 6, 427–434

work page 1928

[11] [11]

Markus Hunziker, Andrei Mart´ ınez-Finkelshtein, Taylor Poe, and Brian Simanek, Poncelet–Darboux, Kippenhahn, and Szeg˝ o: interactions between projective geometry, matrices and orthogonal polynomials, J. Math. Anal. Appl.511(2022), no. 1, Paper No. 126049, 35

work page 2022

[12] [12]

Japan Acad.25(1949), no

Asajiro Ichida,On the foci of algebraic curves, Proc. Japan Acad.25(1949), no. 8, 1–6

work page 1949

[13] [13]

H. M. Jeffrey,On cubics of the third class with triple foci, Quarterly J. Math.14 (1877), 127–139

work page

[14] [14]

,On plane cubics of the third class with three single foci, Quarterly J. Math. 16(1879), 65–81; 348–369

work page

[15] [15]

1-2, 185–225

Rudolf Kippenhahn,On the numerical range of a matrix, Linear Multilinear Algebra 56(2008), no. 1-2, 185–225. Translated from the German by Paul F. Zachlin and Michiel E. Hochstenbach. 14 R. PIENE AND B. SHAPIRO

work page 2008

[16] [16]

Langer and David A

Joel C. Langer and David A. Singer,Foci and foliations of real algebraic curves, Milan J. Math.75(2007), 225–271

work page 2007

[17] [17]

B. Z. Linfield,On certain polar curves with their application to the location of the roots of the derivatives of a rational function, Trans. Amer. Math. Soc.25(1923), no. 2, 239–258

work page 1923

[18] [18]

Reine Angew

Julius Pl¨ ucker, ¨Uber solche Puncte, die bei Curven einer h¨ ohern Ordnung als der zweiten den Brennpuncten der Kegelschnitte entsprechen, J. Reine Angew. Math.10 (1833), 84–91

work page

[19] [19]

Rice,On the foci of plane algebraic curves with applications to symmetric cubic curves, Amer

Ethel A. Rice,On the foci of plane algebraic curves with applications to symmetric cubic curves, Amer. Math. Monthly43(1936), no. 10, 618–630

work page 1936

[20] [20]

R. A. Roberts,On foci and confocal systems of plane curves, Quarterly J. of Pure and Applied Math.32(1901), 141–182

work page 1901

[21] [21]

of Pure and Applied Math

,On foci and confocal plane curves, Quarterly J. of Pure and Applied Math. 35(1904), 297–384

work page 1904

[22] [22]

A treatise on conic sections

George Salmon,A treatise on the higher plane curves: intended as a sequel to “A treatise on conic sections”, Chelsea Publishing Co., New York, 1960. 3rd ed

work page 1960

[23] [23]

Math., vol

Eugenii Shustin,Equiclassical deformation of plane algebraic curves, Singularities (Oberwolfach, 1996), Progr. Math., vol. 162, Birkh¨ auser, Basel, 1998, pp. 195–204

work page 1996

[24] [24]

Siebeck,Ueber eine neue analytische Behandlungsweise der Brennpunkte, J

P. Siebeck,Ueber eine neue analytische Behandlungsweise der Brennpunkte, J. Reine Angew. Math.64(1865), 175–182

work page

[25] [25]

G¨ oschen’sche Verlagshandlung, 1905

Heinrich Wieleitner,Theorie der ebenen Algebraischen Kurven h¨ oherer Ordnung, G.J. G¨ oschen’sche Verlagshandlung, 1905

work page 1905

[26] [26]

Band 61, Springer-Verlag, New York-Heidelberg, 1971

Oscar Zariski,Algebraic surfaces, supplemented edition, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. Band 61, Springer-Verlag, New York-Heidelberg, 1971. With appendices by S. S. Abhyankar, J. Lipman, and D. Mumford. Ragni Piene, Department of Mathematics, University of Oslo, P.O.Box 1053 Blindern, NO-0...

work page 1971