Hyperelliptic Schottky problem solved; AI suggested the final step
Awaiting review
Importance: major (4 of 5)The takeaway
Which abelian varieties are Jacobians of hyperelliptic curves? Huang and Jiang say: exactly those whose theta divisor has singular locus of codimension three. Their acknowledgement says AI tools suggested completing the proof of the main theorem with a known result and constructed the paper’s sharpness example.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 13 November 2026
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None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 134 days before it.
Key facts
- Theorem 1.1: if codim Sing(Θ) = 3 for a ppav (A, Θ), then (A, Θ) ≅ (J(C), Θ_C) for a smooth hyperelliptic curve C; Ein–Lazarsfeld showed codimension ≥ 3 for irreducible Θ, so this characterizes the extreme case, as formulated in Debarre’s note; Beauville proved it for dimension ≤ 5
- Theorem 1.3: for an indecomposable ppav and a coherent ideal sheaf J with J(2Θ) IT₀, X = V(J) is a geometrically nondegenerate GV-subscheme (reduced, irreducible, Cohen–Macaulay, minimal class) — a conjecture of Pareschi and Popa; Meng had the divisor case
- Theorem 1.2 (Debarre’s conjecture): Bs|L| is generically reduced along every codimension-2 component; Example 6.2 (nine non-reduced double points on an abelian threefold) shows codimension 2 is sharp. The abstract says this ‘completes the algebraic-geometry proof by Arbarello-Codogni-Pareschi … of Novikov’s conjecture, which characterizes Jacobians via the KP equation’
- AI disclosure (acknowledgements, verbatim): ‘Besides assisting with editing and revising the manuscript, AI tools suggested us to complete the proof of Theorem 1.1 by applying Theorem 2.8 and constructed the counterexample in Example 6.2. The authors take full responsibility for the mathematical content and the final text.’ Models not named
- Status: arXiv preprint (math.AG), not refereed; no expert reaction found by 13 Nov
What happened
On 11 Nov 2026 Songsong Huang and Zhi Jiang (Fudan) posted a paper characterizing hyperelliptic Jacobians among principally polarized abelian varieties by the size of the singular locus of the theta divisor, a case of the Schottky problem that Beauville had settled only up to dimension five. The key step proves one direction of a conjecture of Pareschi and Popa; a corollary confirms a conjecture of Debarre and, according to the authors, completes the algebraic proof of Novikov’s conjecture by Arbarello, Codogni and Pareschi.
The acknowledgements say that AI tools suggested finishing the proof of the main theorem by applying a result proved earlier in the paper (Theorem 2.8) and constructed the example showing the codimension bound is sharp.
Why it matters
The Schottky problem, recognizing Jacobians among abelian varieties, is one of the classical questions of algebraic geometry; this is a clean criterion for the hyperelliptic case in every dimension. The AI role is stated briefly and the models are not named. The paper is an unrefereed preprint.
What is disputed or not yet verified
| Verification | Preprint |
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Sources
1 source from 1 site. Numbers match the chips in the text.
1 source: 1 primary
Primary
Changes
- Filed (science agent; arXiv PDF read)