arxiv: 2603.04211 · v3 · submitted 2026-03-04 · 🧮 math.AG

Recognition: no theorem link

Planar, rational curves over {mathbb F}₂ whose only singularity is a double point

J\'anos Koll\'ar

Authors on Pith no claims yet

Pith reviewed 2026-05-15 16:18 UTC · model grok-4.3

classification 🧮 math.AG

keywords rational curvesplane curvessingularitiesfinite fieldscharacteristic twoalgebraic geometry

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The pith

Over F_2, planar rational curves exist with a unique double-point singularity in arbitrarily large degree

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

The paper constructs explicit examples of rational curves in the projective plane over the field with two elements whose only singular point is a double point, and these examples have arbitrarily high degree. In characteristic zero the maximum degree for any such curve is six. The constructions therefore demonstrate that the degree restriction disappears once the base field has characteristic two. A reader would care because the examples separate the behavior of rational plane curves by characteristic in a concrete way.

Core claim

We exhibit planar, rational curves of large degree over F_2 that have a unique singular point, which has multiplicity 2.

What carries the argument

Explicit constructions of irreducible plane curves defined over F_2 that possess exactly one singularity of multiplicity two

If this is right

Such curves exist for every sufficiently large degree.
The unique singularity always has multiplicity exactly two.
The curves remain rational and defined over the prime field F_2.

Where Pith is reading between the lines

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

The same pattern may appear for other small characteristics or for curves with slightly worse singularities.
The examples supply test cases for any conjecture that relates rationality, singularities, and the characteristic of the base field.

Load-bearing premise

The explicit constructions produce irreducible curves whose only singularity is a double point of multiplicity exactly two.

What would settle it

Direct computation of the singular locus and factorization for one of the high-degree examples, which would reveal either an extra singularity or reducibility if the claim fails.

read the original abstract

We exhibit planar, rational curves of large degree over ${\mathbb F}_2$ that have a unique singular point, which has multiplicity 2. In characteristic 0 such curves exist only for degrees up to $6$. v.2: references updated and examples of supersingular double planes added. v.3: references updated.

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 claims to exhibit explicit constructions of planar rational curves over the finite field F_2 of arbitrarily large degree that possess exactly one singular point, which is a double point of multiplicity two. This stands in contrast to the characteristic-zero setting, where the authors state that such curves exist only for degrees at most 6.

Significance. If the explicit constructions hold, the result is significant for highlighting characteristic-dependent phenomena in the geometry of rational curves and their singularities. The finitary, algebraic nature of the argument (via direct parametrizations or homogeneous equations over F_2) is a strength, as it permits in-principle computational verification of irreducibility, genus zero, and the precise singularity type without analytic or asymptotic assumptions.

minor comments (2)

[Introduction] The abstract and introduction reference updates in v.2 and v.3 concerning supersingular double planes and references; ensure the main body explicitly incorporates these additions for consistency.
For the general family of constructions, a brief remark on how the chosen homogeneous polynomials guarantee that the image curve is irreducible (rather than a multiple cover) would aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript. The referee's summary correctly identifies the main result: explicit constructions of planar rational curves over F_2 of arbitrarily large degree with exactly one singularity, which is a double point.

Circularity Check

0 steps flagged

No significant circularity; explicit constructions are self-contained

full rationale

The paper's central claim rests on explicit constructions of homogeneous polynomials or parametrizations over F_2. Once the defining equation is given, the required properties (irreducibility, geometric genus zero, unique double point of multiplicity exactly two) reduce to direct algebraic verification of the curve's singularities and parametrization, without any fitted parameters, self-definitional loops, or load-bearing self-citations. No step equates a derived prediction to its own input by construction, and the argument is finitary and algebraic rather than asymptotic or uniqueness-theorem dependent.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on explicit constructions of curves together with standard background results from algebraic geometry over finite fields; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

axioms (1)

standard math Standard results on rational curves, plane curves, and multiplicity of singularities in algebraic geometry.
The paper invokes background theorems on rationality and singularities to certify the constructed objects.

pith-pipeline@v0.9.0 · 5340 in / 1068 out tokens · 54539 ms · 2026-05-15T16:18:22.875914+00:00 · methodology