{"schema":"postcutoff/event@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/e/2026-11-11-hyperelliptic-schottky-problem-huang-jiang-ai/","md":"https://postcutoff.com/e/2026-11-11-hyperelliptic-schottky-problem-huang-jiang-ai/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-11-11-hyperelliptic-schottky-problem-huang-jiang-ai","date":"2026-11-11","date_precision":"day","short_title":"Hyperelliptic Schottky problem solved; AI suggested the final step","deck":null,"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2611.15799: GV-subschemes and hyperelliptic Jacobians (Huang, Jiang)","url":"https://arxiv.org/abs/2611.15799","type":"paper","group":"primary","domain":"arxiv.org"}],"official":1,"filed":"2026-11-13","updated":"2026-11-13","orgs":["Fudan University","various researchers"],"title":"Huang and Jiang claim the hyperelliptic Schottky problem and conjectures of Pareschi–Popa and Debarre; 'AI tools' suggested how to finish the main proof and built the sharpness counterexample","summary":"arXiv 2611.15799 (11 Nov 2026), Songsong Huang and Zhi Jiang: a principally polarized complex abelian variety whose theta divisor has singular locus of codimension three is a hyperelliptic Jacobian (Beauville had this up to dimension five). On the way they prove Pareschi–Popa's Conjecture A direction \"I_X(2Θ) IT₀ ⇒ GV-subscheme\" and, as a corollary, Debarre's conjecture that base schemes of ample line bundles are generically reduced along codimension-2 components, which completes Arbarello–Codogni–Pareschi's algebraic proof of Novikov's KP-equation characterization of Jacobians. Preprint; AI role disclosed in the acknowledgements.","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"],"key_numbers":[],"tags":["math","algebraic-geometry","abelian-varieties","schottky-problem","ai-assisted","arxiv"],"science":{"field":"mathematics","subfield":"algebraic geometry","problem":"Hyperelliptic Schottky problem (codim-3 theta singularities); Pareschi–Popa Conjecture A; Debarre's conjecture on base loci","result":"Codim Sing(Θ) = 3 characterizes hyperelliptic Jacobians; IT₀ criterion for GV-subschemes; generic reducedness of codim-2 base loci","open_since":"","ai_system":["unnamed AI tools"],"human_role":"Human-led; AI suggested the final step of the main proof and constructed the sharpness example","verification":{"key":"preprint","label":"Preprint only","text":"Preprint"},"status":"pending","shock":""},"body_md":"## What happened\n\nOn 11 Nov 2026 Songsong Huang and Zhi Jiang (Fudan) posted a paper characterizing hyperelliptic Jacobians among principally\npolarized abelian varieties by the size of the singular locus of the theta divisor, a case of the Schottky problem that Beauville\nhad settled only up to dimension five. The key step proves one direction of a conjecture of Pareschi and Popa; a corollary confirms\na conjecture of Debarre and, according to the authors, completes the algebraic proof of Novikov's conjecture by Arbarello, Codogni\nand Pareschi.\n\nThe acknowledgements say that AI tools suggested finishing the proof of the main theorem by applying a result proved earlier in\nthe paper (Theorem 2.8) and constructed the example showing the codimension bound is sharp.\n\n## Why it matters\n\nThe Schottky problem, recognizing Jacobians among abelian varieties, is one of the classical questions of algebraic geometry; this\nis a clean criterion for the hyperelliptic case in every dimension. The AI role is stated briefly and the models are not named. The\npaper is an unrefereed preprint.","disputed":[{"label":"Verification","text":"Preprint","sources":[]}],"related":[{"id":"2026-11-13-ai-assisted-results-november-13-listing","url":"https://postcutoff.com/e/2026-11-13-ai-assisted-results-november-13-listing/","date":"2026-11-13","date_precision":"day","short_title":"AI-assisted results, 13 Nov","deck":null,"takeaway":"The arXiv listings of 12–13 Nov carried about 190 maths papers with an AI statement. Most disclose editing or literature search; about forty credit models with proofs, constructions or counterexamples. Seven got their own entries; the rest are catalogued here with the disclosure quoted. All are unrefereed preprints.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":27,"official":27,"filed":"2026-11-13","updated":"2026-11-13","orgs":["OpenAI","Anthropic","Google DeepMind","Axiom Math","various researchers"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-11-13","type":"filed","text":"Created (science agent; arXiv PDF read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-11-13","run":null,"sources_read":"Science agent; arXiv PDF read","updated":"2026-11-13","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":195,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":134,"in_training_data":false},{"model_id":"gemini-4-hydrogen","name":"Gemini 4 Hydrogen","cutoff":"2026-03","days_after":225,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":164,"in_training_data":false}],"short_url":"https://postcutoff.com/s/hyperelliptic-schottky-problem"}