Uniform movable multiples of effective divisors on a fixed surface
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On a fixed smooth projective surface, does every effective divisor D with D²>0 and h⁰(D)=1 have a movable positive multiple bounded solely in terms of the surface?
Why it matters and what counts as progress
Why it matters: A uniform multiple would convert local positivity into a surface-wide bound on when a divisor acquires a moving linear system.
What counts as progress: A uniform bound or divisors on one fixed surface whose required movable multiples grow without bound, with the one-section range proved.
How the problem developed
- 20 Nov 2015 · Open problem [1]
Ciro Ciliberto, Andreas Leopold Knutsen, John Lesieutre, Victor Lozovanu, Rick Miranda, Yusuf Mustopa and Damiano Testa Asked whether positivity hypotheses on an effective divisor D can guarantee an effective or movable multiple whose exponent depends only on the fixed surface.
- 20 May 2026 · Reported claim [2]
Jihao Liu Reported divisors on one fixed smooth projective surface with self-intersection 4 and h⁰(mD)=1 over arbitrarily long ranges, contradicting a uniform multiple bound; the preprint attributes the proof to generative AI, especially ChatGPT 5.5 Pro and Rethlas.
AI contribution
An example of a very non-movable effective divisor
Liu reports effective divisors D_n on one fixed smooth projective surface with D_n²=4 and h⁰(mD_n)=1 for every 1≤m<n²/4+1, for arbitrarily large odd n. Thus no uniform movable multiple depending only on that surface exists.
Sources
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