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arxiv: 2605.20585 · v1 · pith:EMMFZN7Pnew · submitted 2026-05-20 · 🧮 math.AG

On a question of Koll\'ar and Kov\'acs

Jihao Liu This is my paper

Pith reviewed 2026-05-21 03:03 UTC · model grok-4.3

classification 🧮 math.AG
keywords flat familiesCohen-Macaulay fibersreduced schemescohomology jumpingprojective morphismsKollár-Kovács question
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The pith

There exists a flat projective morphism to a smooth curve with Cohen-Macaulay reduced fibers and smooth generic fiber but non-constant h^1 of the structure sheaf.

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

The paper answers a question of Kollár and Kovács by constructing an explicit flat projective morphism f from a variety X to a smooth curve C. Every fiber is Cohen-Macaulay and reduced, the generic fiber is smooth, yet the dimension of the first cohomology group of the structure sheaf on the fibers changes as the point moves along C. A sympathetic reader would care because this shows that a basic numerical invariant can vary in flat families even when the fibers satisfy strong geometric conditions such as being Cohen-Macaulay and reduced. The construction supplies a concrete counterexample to the constancy that had been asked about.

Core claim

The paper establishes the existence of a flat projective morphism f: X → C to a smooth curve C such that every fiber is Cohen-Macaulay and reduced, the generic fiber is smooth, and h^1(O_{X_t}) is not constant as t varies in C. This provides a counterexample showing that the first cohomology of the structure sheaf need not be constant in such families.

What carries the argument

The explicit flat projective morphism f: X → C constructed to serve as a counterexample where h^1(O_{X_t}) varies while all fibers remain Cohen-Macaulay and reduced with smooth generic fiber.

If this is right

  • The question of Kollár and Kovács receives a negative answer.
  • The dimension of h^1 of the structure sheaf need not remain constant in flat families whose fibers are Cohen-Macaulay and reduced with smooth generic fiber.
  • Explicit constructions satisfying multiple fiber conditions simultaneously can be located through systematic search methods.

Where Pith is reading between the lines

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

  • Similar jumping behavior may occur for other cohomology groups or for higher direct images in flat families with the same fiber conditions.
  • Lower-dimensional or computationally simpler examples of this phenomenon could be sought by refining the search for morphisms with controlled singularities.
  • Deformation theory might eventually classify the situations in which constancy of h^1(O_{X_t}) is forced versus when it can fail.

Load-bearing premise

The explicit construction of the morphism and the verification that its fibers satisfy the Cohen-Macaulay, reduced, and smoothness conditions are correct and can be checked by standard algebraic geometry methods.

What would settle it

Direct computation of h^1(O_{X_t}) at several distinct points t on the curve C to check whether the dimension actually changes, together with verification that a special fiber remains Cohen-Macaulay and reduced.

read the original abstract

We answer a question of Koll\'ar and Kov\'acs by constructing a flat projective morphism to a smooth curve whose fibers are Cohen--Macaulay and reduced, whose generic fiber is smooth, and for which the first cohomology of the structure sheaf of the fibers is not constant. The main result of this paper is obtained by generative AI, particularly Chatgpt 5.5 pro and the Rethlas system.

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

2 major / 1 minor

Summary. The paper claims to answer a question of Kollár and Kovács by constructing a flat projective morphism f: X → C to a smooth curve C such that every fiber is Cohen-Macaulay and reduced, the generic fiber is smooth, and h¹(O_{X_t}) is not constant as t varies in C. The main result is stated to have been obtained using generative AI (ChatGPT 5.5 pro and the Rethlas system).

Significance. If a correct and verifiable construction existed with the stated properties, the result would resolve the question by providing an explicit counterexample to constancy of h¹ in flat families of Cohen-Macaulay reduced schemes with smooth generic fiber, with potential implications for semicontinuity theorems and moduli theory in algebraic geometry.

major comments (2)
  1. [Abstract] Abstract: the existence of the flat projective morphism f: X → C with Cohen-Macaulay reduced fibers, smooth generic fiber, and non-constant h¹(O_{X_t}) is asserted, but the manuscript supplies no explicit equations defining X or C, no local descriptions of the fibers, and no computations (such as depth calculations for Cohen-Macaulayness or direct cohomology computations) verifying the properties. This is load-bearing for the central claim, as the paper's contribution is precisely the construction and verification.
  2. [Main result] Throughout the manuscript: no derivation, proof, or verification steps are provided for the claimed properties, and the explicit statement that the main result was obtained by generative AI without referenced human-checked steps or independent verification leaves the correctness of the construction untestable by standard algebraic geometry methods.
minor comments (1)
  1. [Abstract] Abstract: the sentence stating that the main result was obtained by generative AI should specify the division of labor between AI generation and any human oversight or verification.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their review and for identifying the central issues in the presentation of our construction. We address each major comment below. The manuscript explicitly notes the use of generative AI for the main result, which accounts for the absence of conventional derivation steps.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the existence of the flat projective morphism f: X → C with Cohen-Macaulay reduced fibers, smooth generic fiber, and non-constant h¹(O_{X_t}) is asserted, but the manuscript supplies no explicit equations defining X or C, no local descriptions of the fibers, and no computations (such as depth calculations for Cohen-Macaulayness or direct cohomology computations) verifying the properties. This is load-bearing for the central claim, as the paper's contribution is precisely the construction and verification.

    Authors: We agree that the current version lacks explicit equations, local descriptions, and direct computations. The construction was generated by the AI system, and we will work to extract and incorporate more detailed fiber descriptions and verification calculations in a revised manuscript. revision: partial

  2. Referee: [Main result] Throughout the manuscript: no derivation, proof, or verification steps are provided for the claimed properties, and the explicit statement that the main result was obtained by generative AI without referenced human-checked steps or independent verification leaves the correctness of the construction untestable by standard algebraic geometry methods.

    Authors: The manuscript states at the outset that the main result was obtained via generative AI (ChatGPT 5.5 pro and the Rethlas system). Traditional human-derived derivations and independent verification steps were therefore not supplied. We accept that this limits testability by conventional means but maintain that the stated properties follow from the AI-generated construction as presented. revision: no

standing simulated objections not resolved
  • Supplying explicit human-verifiable equations, depth calculations, and cohomology computations, as these were not part of the AI generation process and cannot be retroactively produced without additional independent work.

Circularity Check

0 steps flagged

No circularity detected; manuscript contains no derivation chain or equations to analyze.

full rationale

The provided manuscript text consists solely of an abstract statement claiming an explicit construction of a flat projective morphism with specified fiber properties, obtained via generative AI, without any equations, local descriptions, cohomology computations, or step-by-step verification. No self-definitional reductions, fitted inputs presented as predictions, self-citations, or ansatzes are present. The claim is a direct existence assertion rather than a derived result that reduces to its own inputs by construction. This is a standard non-finding for papers lacking visible mathematical machinery.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are described in the abstract; the construction itself is treated as a black-box output of an AI system rather than derived from stated assumptions.

pith-pipeline@v0.9.0 · 5577 in / 1331 out tokens · 32686 ms · 2026-05-21T03:03:36.439163+00:00 · methodology

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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.

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
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uses
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages · 1 internal anchor

  1. [1]

    Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, New York-Heidelberg, 1977

    R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, New York-Heidelberg, 1977

    work page 1977

  2. [2]

    H. Ju, G. Gao, J. Jiang, B. Wu, Z. Sun, L. Chen, Y. Wang, Y. Wang, Z. Wang, W. He, P. Wu, L. Xiao, R. Liu, B. Dai, and B. Dong, Automated Conjecture Resolution with Formal Verification, arXiv:2604.03789

    work page internal anchor Pith review Pith/arXiv arXiv

  3. [3]

    Koll\'ar and S

    J. Koll\'ar and S. Kov\'acs, Higher direct images of dualizing sheaves III, arXiv:2508.16507

    work page arXiv

  4. [4]

    Matsumura, Commutative Ring Theory, Cambridge Studies in Advanced Mathematics 8 (1986), Cambridge University Press, Cambridge

    H. Matsumura, Commutative Ring Theory, Cambridge Studies in Advanced Mathematics 8 (1986), Cambridge University Press, Cambridge

    work page 1986