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arxiv: 2311.00963 · v2 · submitted 2023-11-02 · 🧮 math.AG

Log canonical thresholds of high multiplicity reduced plane curves

Erik Paemurru , Nivedita Viswanathan This is my paper

Pith reviewed 2026-05-24 06:19 UTC · model grok-4.3

classification 🧮 math.AG
keywords log canonical thresholdplane curvemultiplicitysingularitiesalgebraic geometryreduced curvedegree d curve
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The pith

Log canonical thresholds of reduced plane curves of degree d are computed at points of multiplicity d-1.

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

The paper computes the log canonical thresholds of reduced plane curves of degree d at points of multiplicity d-1. This yields a complete description of every possible log canonical threshold value less than 2/(d-1) that can occur for any reduced plane curve of degree d. The authors also give explicit values for all reduced plane curves of degree less than 6. These results supply concrete data on the local singularities of such curves.

Core claim

We compute log canonical thresholds of reduced plane curves of degree d at points of multiplicity d-1. As a consequence, we describe all possible values of log canonical threshold that are less than 2/(d-1) for reduced plane curves of degree d. In addition, we compute log canonical thresholds for all reduced plane curves of degree less than 6.

What carries the argument

The log canonical threshold of the curve at the point of multiplicity d-1, obtained through explicit local computations.

If this is right

  • All possible log canonical threshold values below 2/(d-1) are now listed for reduced plane curves of any degree d.
  • Explicit log canonical threshold values are available for every reduced plane curve of degree at most 5.
  • The range of attainable singularities for these curves is now delimited by the computed thresholds.

Where Pith is reading between the lines

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

  • The same local methods could be tested on curves whose maximum multiplicity is slightly less than d-1 to see whether similar closed-form expressions appear.
  • The explicit list of small thresholds may be compared against existing databases of plane curve singularities to check consistency with other invariants such as the Milnor number.
  • The results supply concrete test cases for any proposed general formula that bounds or predicts log canonical thresholds of plane curves.

Load-bearing premise

The base field is algebraically closed of characteristic zero and the curve is reduced with a point of exact multiplicity d-1.

What would settle it

A reduced plane curve of degree d over an algebraically closed field of characteristic zero with a point of multiplicity exactly d-1 whose log canonical threshold differs from the computed value.

read the original abstract

We compute log canonical thresholds of reduced plane curves of degree $d$ at points of multiplicity $d-1$. As a consequence, we describe all possible values of log canonical threshold that are less than $2/(d-1)$ for reduced plane curves of degree $d$. In addition, we compute log canonical thresholds for all reduced plane curves of degree less than 6.

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 / 2 minor

Summary. The paper computes the log canonical thresholds of reduced plane curves of degree d at points of multiplicity exactly d-1. It uses the standard discrepancy formula after successive blow-ups at the point and its infinitely near points. As a consequence, it describes all possible lct values less than 2/(d-1) for reduced degree-d plane curves, and provides explicit computations for all reduced plane curves of degree less than 6.

Significance. If the computations hold, the result gives explicit lct values in the high-multiplicity regime where the degree bound truncates the possible resolution graphs, enabling exhaustive case analysis on weighted homogeneous initial forms of degree d-1 followed by at most one further blow-up. The low-degree tables serve as an internal consistency check, and the consequence that every lct < 2/(d-1) must arise from multiplicity exactly d follows from the inequality lct(f) ≥ 2/m for m ≤ d-2 together with the degree bound.

minor comments (2)
  1. The abstract and introduction could more explicitly reference the precise statement of the discrepancy formula used in the blow-up computations (e.g., the formula for a(E,X,0) in terms of the orders after each blow-up).
  2. For the low-degree tables (d < 6), it would help to indicate whether the listed lct values are achieved only at multiplicity d-1 or also at lower multiplicities, to clarify the scope of the consequence.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; computations are self-contained via standard resolution

full rationale

The paper computes lct values for reduced plane curves of degree d with multiplicity d-1 via explicit successive blow-ups at the point and infinitely near points, applying the standard discrepancy formula. The degree-d bound truncates the possible initial forms and resolution graphs, permitting exhaustive case analysis without any fitted parameters, self-definitional reductions, or load-bearing self-citations. The consequence that lct < 2/(d-1) implies multiplicity exactly d-1 follows directly from the inequality lct(f) ≥ 2/m for m ≤ d-2 together with the degree truncation, both verified by the same bookkeeping. Low-degree tables (d < 6) serve as internal consistency checks. No step reduces by construction to its inputs or to prior author work invoked as a uniqueness theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Paper relies on standard domain assumptions of algebraic geometry; no free parameters or invented entities visible from abstract.

axioms (1)
  • domain assumption Base field is algebraically closed of characteristic zero.
    Standard background assumption for resolution of singularities and LCT computations in algebraic geometry.

pith-pipeline@v0.9.0 · 5577 in / 1099 out tokens · 28706 ms · 2026-05-24T06:19:47.316613+00:00 · methodology

discussion (0)

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Reference graph

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