---
id: "2026-11-11-hyperelliptic-schottky-problem-huang-jiang-ai"
url: "https://postcutoff.com/e/2026-11-11-hyperelliptic-schottky-problem-huang-jiang-ai/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-11-11"
date_precision: day
category: science
importance: 4
confidence: medium
status: [Awaiting review]
verification: Preprint only
sources: 1
editor: human editor
human_review: null
version: null
---

As of: 2026-11-13 13:49 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-11-11-hyperelliptic-schottky-problem-huang-jiang-ai/

# Hyperelliptic Schottky problem solved; AI suggested the final step

Full 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

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

## 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

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 195 days after its cutoff
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- Grok 4.7 (training cutoff May 2026): 164 days after its cutoff

## Sources

1. [arXiv 2611.15799: GV-subschemes and hyperelliptic Jacobians (Huang, Jiang)](https://arxiv.org/abs/2611.15799) (arxiv.org, paper)

## Changes

- 2026-11-13 (filed): Created (science agent; arXiv PDF read)

## Related

- 2026-11-13: [AI-assisted results, 13 Nov](https://postcutoff.com/e/2026-11-13-ai-assisted-results-november-13-listing/index.md)