An example of a very non-movable effective divisor

Jihao Liu

arxiv: 2605.20594 · v1 · pith:UVPEFC3Nnew · submitted 2026-05-20 · 🧮 math.AG

An example of a very non-movable effective divisor

Jihao Liu This is my paper

Pith reviewed 2026-05-21 02:58 UTC · model grok-4.3

classification 🧮 math.AG

keywords effective divisorself-intersectionnon-movable divisorprojective surfacecounterexamplealgebraic geometry

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The pith

There exists a very non-movable effective divisor of positive self-intersection on a smooth projective surface.

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

This paper establishes the existence of an effective divisor on a smooth projective surface that has positive self-intersection but is very non-movable. It thereby supplies a negative answer to a question of Ciliberto and collaborators on whether positive self-intersection forces movability for effective divisors. A sympathetic reader would care because the example separates self-intersection positivity from movability in a concrete way. The result rests on an explicit construction rather than a general theorem.

Core claim

The paper exhibits a specific effective divisor on a smooth projective surface that has positive self-intersection yet fails to be movable in the strong sense required by the question, directly supplying a negative answer to the problem posed by Ciliberto, Knutsen, Lesieutre, Lozovanu, Miranda, Mustopa, and Testa.

What carries the argument

The very non-movable effective divisor, which is built so that its self-intersection is positive while its linear system remains fixed in a way that prevents movability.

If this is right

Positive self-intersection no longer guarantees that an effective divisor is movable on smooth projective surfaces.
The distinction between movable and non-movable divisors must be tracked separately from the sign of the self-intersection form.
Questions about the geometry of linear systems on surfaces require finer invariants than self-intersection alone.

Where Pith is reading between the lines

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

Similar examples might exist on surfaces with other invariants or in higher-dimensional varieties.
The construction could be adapted to test related questions about the movable cone versus the effective cone.
Direct verification of the numerical data in the example would allow the result to be used in further classifications of surface divisors.

Load-bearing premise

The explicit construction produces a divisor whose intersection numbers are positive and whose non-movability properties hold as stated.

What would settle it

An independent calculation of the divisor's self-intersection number together with a check that its linear system has the dimension and base locus required for very non-movability.

read the original abstract

We give a negative answer to a question of Ciliberto, Knutsen, Lesieutre, Lozovanu, Miranda, Mustopa, and Testa on effective divisors of positive self-intersection on smooth projective surfaces. 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.

Desk Editor's Note private letter to a colleague

This paper claims to give a negative answer to a question on non-movable divisors via an AI-generated example, but supplies no explicit calculations or proof steps to support it.

read the letter

Colleague, The one or two things to know: this paper gives a negative answer to the question posed by Ciliberto, Knutsen and the others about whether there exists a very non-movable effective divisor with positive self-intersection on a smooth projective surface. It does so by constructing an example. On the positive side, if the example holds, it directly addresses an open question in the literature on divisors and their properties. The paper is upfront that the construction came from using ChatGPT 5.5 pro and the Rethlas system, which at least makes the origin clear. However, the main issue is that no explicit details are provided for how the surface is defined, what the divisor is, how the self-intersection is calculated to be positive, or how the non-movability is established. These are the core parts that need to be checked in algebraic geometry. Relying on AI output without showing the verification steps makes it difficult to assess if everything is correct or if there are errors in the generation. The stress test note points out exactly this gap: no human-readable derivation or computations. That seems accurate based on what's described. This kind of work would appeal to researchers focused on questions about movable divisors and surfaces in algebraic geometry. But for it to be useful, the example needs to be independently verifiable, which it isn't in the current presentation. Overall, I think this does not deserve to go to peer review yet. The authors should first provide the full mathematical details and perhaps have the construction checked by hand or with computer algebra software to confirm the properties.

Referee Report

2 major / 0 minor

Summary. The paper claims to provide a negative answer to a question of Ciliberto et al. by exhibiting an example of a very non-movable effective divisor D with D² > 0 on a smooth projective surface X. The construction is stated to have been obtained via generative AI (ChatGPT 5.5 pro and the Rethlas system).

Significance. If the example were independently verified with explicit equations and checks, it would resolve the question negatively and contribute a concrete counterexample to the study of movable divisors on surfaces.

major comments (2)

The manuscript provides no explicit description of the surface X, the class of the divisor D, the computation of the self-intersection D², or the verification that |D| satisfies the very non-movable criterion of Ciliberto et al. These numerical and cohomological checks are load-bearing for the central claim but are absent from the text.
The abstract states that the main result is obtained by generative AI with no indication of subsequent human verification, detailed proof steps, or independent error checking. In algebraic geometry, such an example requires reproducible calculations to be accepted as a valid counterexample.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and the detailed report. We respond point by point to the major comments below.

read point-by-point responses

Referee: The manuscript provides no explicit description of the surface X, the class of the divisor D, the computation of the self-intersection D², or the verification that |D| satisfies the very non-movable criterion of Ciliberto et al. These numerical and cohomological checks are load-bearing for the central claim but are absent from the text.

Authors: We agree that the submitted manuscript lacks these explicit details and computations. The construction originated from the generative AI system, and the initial output did not include the supporting equations or verifications. We will revise the paper to incorporate a concrete description of the surface X, the divisor class, the explicit computation of D² > 0, and the cohomological checks confirming the very non-movable property according to the criteria of Ciliberto et al. revision: yes
Referee: The abstract states that the main result is obtained by generative AI with no indication of subsequent human verification, detailed proof steps, or independent error checking. In algebraic geometry, such an example requires reproducible calculations to be accepted as a valid counterexample.

Authors: The abstract already discloses the use of generative AI. We have carried out subsequent human review and independent checks of the construction. The revised version will include an expanded account of these verification steps, with detailed calculations and error-checking procedures, to make the example fully reproducible. revision: partial

Circularity Check

0 steps flagged

No circularity: result presented as external AI-generated example with no internal derivation chain.

full rationale

The paper states that its main result—an explicit example of a very non-movable effective divisor of positive self-intersection—is obtained directly from generative AI (ChatGPT 5.5 pro and the Rethlas system). No mathematical derivation, sequence of equations, or first-principles argument is supplied in the abstract or described text that could reduce to its own inputs by construction. The claim is framed as an external construction answering a question posed by Ciliberto et al., rather than a self-referential computation or fitted prediction. While the absence of explicit intersection calculations or non-movability verification raises separate verifiability concerns, these do not constitute circularity under the defined patterns, as there is no load-bearing step that equates output to input via definition, self-citation, or renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper operates inside the standard framework of algebraic geometry on smooth projective surfaces; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

standard math Standard definitions of effective divisors, self-intersection numbers, and movability on smooth projective surfaces over the complex numbers.
The result is stated in the language of classical algebraic geometry without redefining core notions.

pith-pipeline@v0.9.0 · 5566 in / 1126 out tokens · 46364 ms · 2026-05-21T02:58:46.420195+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages · 1 internal anchor

[1]

Ciliberto, A

C. Ciliberto, A. L. Knutsen, J. Lesieutre, V. Lozovanu, R. Miranda, Y. Mustopa, and D. Testa, A few questions about curves on surfaces, Rend. Circ. Mat. Palermo, II. Ser. 66 (2017), 195--204

work page 2017
[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

[1] [1]

Ciliberto, A

C. Ciliberto, A. L. Knutsen, J. Lesieutre, V. Lozovanu, R. Miranda, Y. Mustopa, and D. Testa, A few questions about curves on surfaces, Rend. Circ. Mat. Palermo, II. Ser. 66 (2017), 195--204

work page 2017

[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