Singular Points of High Multiplicity for Septic Curves

David A. Weinberg; Nicholas J. Willis

arxiv: 1906.11214 · v1 · pith:ONIEP4PKnew · submitted 2019-06-26 · 🧮 math.AG

Singular Points of High Multiplicity for Septic Curves

Nicholas J. Willis , David A. Weinberg This is my paper

Pith reviewed 2026-05-25 15:00 UTC · model grok-4.3

classification 🧮 math.AG

keywords singular pointsmultiplicityseptic curvesplane algebraic curvesreal curvescomplex curvessingularity classificationirreducible curves

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The pith

Real irreducible septic curves have 22 types of multiplicity-six singular points and 174 of multiplicity five.

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

The paper classifies singular points of multiplicity four through six on irreducible plane algebraic curves of degree seven. It gives exact counts of distinct types over the reals and over the complexes for the two highest multiplicities, along with lower bounds for multiplicity four. A reader would care because these counts inventory the possible local structures near the most constrained singular points on such curves. The work concludes by listing explicit open problems in the broader classification of singular points on plane curves.

Core claim

For real irreducible algebraic curves of the seventh degree, there are 22 types of singular points of multiplicity six, 174 types of singular points of multiplicity five, and at least 182 types of singular points of multiplicity four. For complex irreducible algebraic curves of the seventh degree, there are 12 types of singular points of multiplicity six, 92 types of singular points of multiplicity five, and at least 92 types of singular points of multiplicity four.

What carries the argument

Enumeration of inequivalent local analytic or topological structures of singular points, distinguished under multiplicity and global irreducibility constraints for degree-seven curves.

If this is right

The counts for multiplicities five and six are presented as complete.
Multiplicity-four counts are given only as lower bounds, indicating the enumeration is partial.
The final section explicitly describes open problems on classifying singular points of plane algebraic curves of various degrees and multiplicities.

Where Pith is reading between the lines

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

The real-versus-complex distinction highlights how reality constraints increase the number of distinct types.
The methods could extend to producing similar inventories for curves of degree eight or nine with high-multiplicity points.

Load-bearing premise

The listed types form an exhaustive partition of all possible inequivalent local structures without omissions or duplicates.

What would settle it

A real irreducible septic curve whose multiplicity-six singular point has a local structure matching none of the 22 enumerated types.

read the original abstract

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 supplies new tabulated counts of singularity types for multiplicities 4-6 on irreducible septic curves in both real and complex cases, but the abstract gives no method or verification details.

read the letter

The key takeaway is that this paper gives concrete counts for the number of inequivalent singular point types of multiplicity 4, 5, and 6 on irreducible plane curves of degree 7, separating the real and complex cases. For real curves the numbers are 22 for multiplicity 6, 174 for 5, and at least 182 for 4; for complex they are 12, 92, and at least 92 respectively. These appear to be fresh numerical results within the existing classification program for plane curve singularities. The final section that lists open problems is a straightforward addition that flags remaining gaps for other degrees and multiplicities. The work stays tightly focused on supplying these specific tallies rather than developing new invariants or techniques. The main limitation is that the abstract states the counts without describing the classification procedure, the local invariants used to separate types, or any steps taken to confirm the lists are complete and free of duplicates. This makes it impossible to judge accuracy from the given text alone. The qualifier “at least” for multiplicity 4 already shows the enumeration is known to be partial there, which is honest but narrows the strength of that claim. The paper is aimed at the small group of algebraic geometers who work on explicit classification of singularities on curves of fixed low degree. A reader already engaged in that sub-area could use the numbers as reference data, but the piece does not reorganize the field or supply tools that would be adopted more broadly. I would send it to peer review so that specialists can inspect the full classification arguments and check whether the distinctions hold up.

Referee Report

1 major / 0 minor

Summary. The manuscript enumerates the distinct types of singular points of multiplicities 6, 5, and 4 on irreducible plane algebraic curves of degree 7. It reports 22 (real) / 12 (complex) types for multiplicity 6, 174 / 92 types for multiplicity 5, and at least 182 / 92 types for multiplicity 4, while listing a variety of open problems on the classification of singular points of plane algebraic curves in its final section.

Significance. If the enumerations prove exhaustive and the inequivalent local analytic or topological structures are correctly distinguished under the irreducibility constraints, the explicit counts would constitute a concrete contribution to singularity classification for septic curves, a setting where high-multiplicity cases are combinatorially intricate. The inclusion of both real and complex cases together with an explicit list of open problems would help delineate the current boundaries of the classification problem.

major comments (1)

[Abstract] Abstract: the central counts (22/12, 174/92, at least 182/92) are asserted without any description of the classification method, the invariants used to distinguish types, the verification procedure, or error bounds on exhaustiveness. This absence renders the soundness of the load-bearing claim—that the listed numbers are complete and correctly separate inequivalent structures—impossible to assess from the supplied text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and the opportunity to respond. The single major comment concerns the abstract; we address it directly below and agree that a revision is warranted.

read point-by-point responses

Referee: [Abstract] Abstract: the central counts (22/12, 174/92, at least 182/92) are asserted without any description of the classification method, the invariants used to distinguish types, the verification procedure, or error bounds on exhaustiveness. This absence renders the soundness of the load-bearing claim—that the listed numbers are complete and correctly separate inequivalent structures—impossible to assess from the supplied text.

Authors: We agree that the abstract, in its current form, provides no indication of the classification method, the distinguishing invariants, the verification steps, or the status of exhaustiveness. The abstract was written for brevity, but this omission makes the central claims difficult to evaluate on their own. In the revised manuscript we will expand the abstract by one or two sentences that state: the enumeration proceeds by exhaustive case-by-case analysis of possible local analytic and topological types (distinguished by the tangent cone, the sequence of multiplicities along the branches, and the resolution graph) that can be realized by an irreducible plane curve of degree 7; verification combines direct algebraic constructions with computer-assisted checks of the relevant Hilbert-Samuel functions and intersection multiplicities; and the counts for multiplicities 6 and 5 are claimed to be complete while the multiplicity-4 count is presented only as a lower bound. The full technical details, including the precise invariants and the arguments for completeness or incompleteness, remain in Sections 2–4 of the paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper enumerates singularity types for irreducible plane septic curves by multiplicity (4,5,6) over reals and complexes, relying on case-by-case distinction of local analytic or topological invariants. The abstract and description contain no equations, parameter fits, self-citations, or ansatzes that reduce the claimed counts to inputs by construction. No load-bearing step is self-definitional or renames a known result as a derivation; the result is presented as exhaustive classification independent of the final tallies.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; ledger left empty.

pith-pipeline@v0.9.0 · 5618 in / 1046 out tokens · 19059 ms · 2026-05-25T15:00:08.930488+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Walker, Algebraic Curves, Princeton University Pr ess, New Jer- sey, 1950, 2nd ed

R.J. Walker, Algebraic Curves, Princeton University Pr ess, New Jer- sey, 1950, 2nd ed. published by Springer-Verlag, New York, 1 978

work page 1950
[2]

Wall, Singular points of plane curves, London Mat hematical Society Student Texts, Vol

C.T.C. Wall, Singular points of plane curves, London Mat hematical Society Student Texts, Vol. 63, Cambridge University Press , 2004

work page 2004
[3]

Weinberg and Nicholas J

David A. Weinberg and Nicholas J. Willis, Singular point s of real quartic and quintic curves, Tbilisi Mathematical Journal, Vol. 2 (2009), pp. 95 - 134

work page 2009
[4]

Weinberg and Nicholas J

David A. Weinberg and Nicholas J. Willis, Singular point s of real sex- tic curves I, Acta Applicandae Mathematicae, Vol. 110, no. 2 (2010), pp. 805 - 862

work page 2010
[5]

Weinberg and Nicholas J

David A. Weinberg and Nicholas J. Willis, Singular point s of re- ducible sextic curves, ISRN Geometry, Vol. 2012 (2012), Art icle ID 680247, 17 pages (www.hindawi.com/isrn/geometry/2012/6 80247/). David A. Weinberg: Department of Mathematics and Statis- tics, Texas Tech University, Lubbock, Texas 79409-1042 E-mail address: david.weinberg@ttu.edu Nicholas...

work page 2012

[1] [1]

Walker, Algebraic Curves, Princeton University Pr ess, New Jer- sey, 1950, 2nd ed

R.J. Walker, Algebraic Curves, Princeton University Pr ess, New Jer- sey, 1950, 2nd ed. published by Springer-Verlag, New York, 1 978

work page 1950

[2] [2]

Wall, Singular points of plane curves, London Mat hematical Society Student Texts, Vol

C.T.C. Wall, Singular points of plane curves, London Mat hematical Society Student Texts, Vol. 63, Cambridge University Press , 2004

work page 2004

[3] [3]

Weinberg and Nicholas J

David A. Weinberg and Nicholas J. Willis, Singular point s of real quartic and quintic curves, Tbilisi Mathematical Journal, Vol. 2 (2009), pp. 95 - 134

work page 2009

[4] [4]

Weinberg and Nicholas J

David A. Weinberg and Nicholas J. Willis, Singular point s of real sex- tic curves I, Acta Applicandae Mathematicae, Vol. 110, no. 2 (2010), pp. 805 - 862

work page 2010

[5] [5]

Weinberg and Nicholas J

David A. Weinberg and Nicholas J. Willis, Singular point s of re- ducible sextic curves, ISRN Geometry, Vol. 2012 (2012), Art icle ID 680247, 17 pages (www.hindawi.com/isrn/geometry/2012/6 80247/). David A. Weinberg: Department of Mathematics and Statis- tics, Texas Tech University, Lubbock, Texas 79409-1042 E-mail address: david.weinberg@ttu.edu Nicholas...

work page 2012