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arxiv: 2606.26944 · v1 · pith:GLVNILXQnew · submitted 2026-06-25 · 🧮 math.AG · math.CV

On a counterexample to a conjecture of J. Harris for octic surfaces

Hossein Movasati This is my paper

Pith reviewed 2026-06-26 03:04 UTC · model grok-4.3

classification 🧮 math.AG math.CV
keywords Noether-Lefschetz locioctic surfacesFermat surfaceHarris conjecturecodimensioncohomology classesalgebraic curvescomplete intersection
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The pith

For all but finitely many rational r, the Noether-Lefschetz loci of C1 + r C2 on the octic Fermat surface are distinct 31-codimensional subvarieties.

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

The paper studies linear combinations C1 + r C2, where C1 is a line and C2 is a (3,3)-complete intersection curve on the Fermat octic surface, chosen so that the two curves meet nowhere. Strong computational evidence is assembled to show that the Noether-Lefschetz loci attached to the cohomology classes of these curves remain set-theoretically distinct 31-codimensional subvarieties for every rational r outside a finite list. These loci meet pairwise in 32-codimensional subvarieties, and the highest codimension attained by any component reaches 35. The construction is offered as a possible counterexample to a conjecture of J. Harris on the geometry of such loci.

Core claim

For all except a finite number of r, the Noether-Lefschetz loci attached to the cohomology classes of C1 + r C2 are set theoretically distinct 31 codimensional subvarieties intersecting each other in a 32 codimensional subvariety of the ambient space. The maximum codimension for components of the Noether-Lefschetz locus in this case is 35.

What carries the argument

The Noether-Lefschetz loci attached to the cohomology classes of the curves C1 + r C2.

If this is right

  • The Noether-Lefschetz loci attached to distinct r remain set-theoretically distinct outside a finite set.
  • Any two such loci intersect in a 32-codimensional subvariety.
  • Components of the Noether-Lefschetz locus reach a maximum codimension of 35.
  • The construction supplies a candidate counterexample to Harris's conjecture on the codimensions of these loci.

Where Pith is reading between the lines

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

  • The same linear-combination technique with disjoint curves may distinguish loci on other Fermat hypersurfaces of even degree.
  • The finite list of exceptional r is likely determined by arithmetic conditions on the periods of the curves.
  • If the pattern persists, the full Noether-Lefschetz locus on the octic would contain a 32-dimensional skeleton formed by these intersections.

Load-bearing premise

That the gathered evidences suffice to establish the loci are distinct and of the stated codimensions for all but finitely many r, without post-hoc selection of r values or hidden dependencies in the cohomology class computations.

What would settle it

An explicit computation for some large rational r showing that two different values produce the same locus or that the codimension differs from 31 would falsify the claim.

read the original abstract

We take a sum $C_1+r C_2,\ r\in\Q$ of a line $C_1$ and a complete intersection curve $C_2$ of type $(3,3)$ inside the octic Fermat surface and with no intersection points. We gather strong evidences to the fact that for all except a finite number of $r$, the Noether-Lefschetz loci attached to the cohomology classes of $C_1+r C_2$ are set theoretically distinct $31$ codimensional subvarieties intersecting each other in a $32$ codimensional subvariety of the ambient space. The maximum codimension for components of the Noether-Lefschetz locus in this case is $35$, and hence, we provide a possible counterexample to a conjecture of J. Harris.

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

3 major / 2 minor

Summary. The manuscript studies Noether-Lefschetz loci in the moduli space of octic surfaces associated to the cohomology classes of curves of the form C1 + r C2, where C1 is a line and C2 is a (3,3)-complete intersection curve on the Fermat octic surface with empty intersection. Using computational evidence for rational values of r, the authors claim that for all but finitely many such r the corresponding NL loci are distinct 31-codimensional subvarieties whose pairwise intersections are 32-codimensional, with the maximum codimension of any component being 35; this is offered as a possible counterexample to a conjecture of J. Harris.

Significance. If the observed codimensions and distinctness can be established rigorously for all but finitely many r, the result would supply a concrete counterexample to Harris' conjecture on the expected dimensions of Noether-Lefschetz loci, which remains a topic of interest in the Hodge theory of surfaces. The use of rational linear combinations of geometrically independent curves to produce families of distinct loci is a potentially useful technique, though its generality is not yet demonstrated.

major comments (3)
  1. [Abstract and §1] Abstract and §1: The central assertion that the NL loci attached to [C1 + r C2] are distinct 31-codimensional subvarieties for all but finitely many rational r is supported only by computational evidence on a finite collection of r values. No uniformity theorem, deformation argument, or asymptotic control on the rank of the Picard lattice or the primitivity of the class is supplied to justify that the observed codimensions persist outside this finite set, leaving open the possibility of r-dependent relations in the cohomology ring that could alter the codimension for infinitely many r.
  2. [§4] §4 (computational evidence): The codimension computations and distinctness statements are presented for selected r without an explicit list of the tested values, the algorithm used to compute the relevant Hodge classes or the dimension of the loci, or a verification that the chosen r avoid accidental linear dependencies. This makes it impossible to assess whether the finite-exception clause is an artifact of post-hoc selection.
  3. [§3] §3 (main claim): The statement that the maximum codimension of components is 35 and that pairwise intersections are exactly 32-codimensional is asserted on the basis of the same finite computational sample; without a proof that these dimensions are stable under specialization or deformation in the parameter r, the numerical values remain conditional on the specific r examined.
minor comments (2)
  1. [Abstract] The abstract refers to 'strong evidences' without specifying the number of r tested or the software/algorithm employed; a brief description of the computational method would improve reproducibility.
  2. [§2] Notation for the ambient moduli space and the precise definition of the Noether-Lefschetz locus (e.g., as a subscheme of the moduli space of octics) should be introduced earlier and used consistently.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript, which presents computational evidence toward a possible counterexample to Harris's conjecture. We respond point by point below, clarifying the evidential nature of the claims and indicating revisions where details can be improved.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: The central assertion that the NL loci attached to [C1 + r C2] are distinct 31-codimensional subvarieties for all but finitely many rational r is supported only by computational evidence on a finite collection of r values. No uniformity theorem, deformation argument, or asymptotic control on the rank of the Picard lattice or the primitivity of the class is supplied to justify that the observed codimensions persist outside this finite set, leaving open the possibility of r-dependent relations in the cohomology ring that could alter the codimension for infinitely many r.

    Authors: We agree that the central claims rest on computational evidence for finitely many rational r and that no uniformity theorem or deformation argument is provided. The manuscript is framed as gathering 'strong evidences' for the stated properties holding for all but finitely many r, without claiming a proof. We will revise the abstract and §1 to state more explicitly that the distinctness, 31-codimensionality, and finite-exception clause are observed in the tested cases and conjectured to persist more generally on the basis of this evidence. revision: partial

  2. Referee: [§4] §4 (computational evidence): The codimension computations and distinctness statements are presented for selected r without an explicit list of the tested values, the algorithm used to compute the relevant Hodge classes or the dimension of the loci, or a verification that the chosen r avoid accidental linear dependencies. This makes it impossible to assess whether the finite-exception clause is an artifact of post-hoc selection.

    Authors: We acknowledge that §4 lacks sufficient detail on the tested values and methods. In the revised manuscript we will expand §4 to include an explicit list of the rational r examined, a description of the algorithm (based on explicit computation of the relevant Hodge classes via the cohomology ring of the Fermat octic and dimension calculations in the moduli space), and checks confirming linear independence of the classes for those r. revision: yes

  3. Referee: [§3] §3 (main claim): The statement that the maximum codimension of components is 35 and that pairwise intersections are exactly 32-codimensional is asserted on the basis of the same finite computational sample; without a proof that these dimensions are stable under specialization or deformation in the parameter r, the numerical values remain conditional on the specific r examined.

    Authors: The statements in §3 concerning the maximum codimension of 35 and 32-codimensional intersections are indeed drawn from the finite computational sample. We do not possess a proof of stability under specialization or deformation in r. We will revise §3 to present these as observed values across the tested r, thereby supplying evidence for the claimed dimensions while qualifying their conditional status. revision: partial

standing simulated objections not resolved
  • We are unable to supply a uniformity theorem, deformation argument, or proof that the observed codimensions and distinctness hold for all but finitely many r, as this lies outside the scope of the computational evidence presented.

Circularity Check

0 steps flagged

No circularity; paper presents computational evidence for a conjecture rather than a closed derivation

full rationale

The manuscript states that it 'gather[s] strong evidences' for the distinctness and codimensions of Noether-Lefschetz loci attached to classes [C1 + r C2] for all but finitely many rational r. No derivation chain, uniqueness theorem, ansatz, or parameter-fitting step is described that reduces the claimed statements to their own inputs by construction. The central assertion remains an evidence-based claim whose validity can be checked externally via independent computation of Hodge classes and Picard ranks, with no self-citation load-bearing or self-definitional reduction present.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are described.

pith-pipeline@v0.9.1-grok · 5666 in / 1170 out tokens · 70205 ms · 2026-06-26T03:04:37.710007+00:00 · methodology

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Reference graph

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