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The ordinary-double-point gap in every dimension
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Abstract
We prove the ordinary-double-point gap conjecture for boundary-zero complex algebraic klt germs in every dimension: a singular n-dimensional germ has normalized volume at most $2(n-1)^n$, with equality precisely at an analytic ordinary double point.
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Related prior literature mentioned by others
- Cristiano Spotti and Song Sun, Explicit Gromov–Hausdorff compactifications of moduli spaces of Kähler–Einstein Fano manifolds (arXiv:1705.00377, 2017). The AI Proved What? family exposition identifies this work as the source of the ordinary-double-point volume-gap conjecture addressed by both manuscripts. [family-level]
- Chi Li, Minimizing normalized volumes of valuations (arXiv:1511.08164, 2015; revised 2017). The AI Proved What? family exposition cites this work for the normalized-volume invariant used in both manuscripts. [family-level]
- Yuchen Liu, K-stability of cubic fourfolds (arXiv:2007.14320, 2020; revised 2022). The AI Proved What? family exposition identifies Liu's local-complete-intersection case as prior work, and the OpenAI Math Digest fourfold guide explains its use after the claimed embedding-dimension reduction. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Chenyang Xu and Ziquan Zhuang, Open problems in K-stability of Fano varieties (arXiv:2601.15576, v2 1 June 2026). Lists the conjecture this paper's Theorem 1.1 addresses as Problem 33 (Section 3.2) and says an affirmative answer would imply that the K-moduli of cubic hypersurfaces agrees with the GIT moduli; it is not in this paper's bibliography, although the dimension-four companion cites it.
- Zhiyu Liu, The finite degree formula for normalized volumes (arXiv:2607.27032, July 2026). Proves a finite degree formula for finite surjective crepant morphisms of klt singularities, which relates to the paper's use of the finite degree formula (cited from Xu–Zhuang) to get Gorenstein structure on the K-polystable cone.
Digests and expositions
- OpenAI Math Digest, The ordinary-double-point gap in every dimension (online exposition, undated). This guide maps the proposed proof through stable degeneration, jet-semigroup saturation, the same-dimensional supremum argument, and the global Fano consequences, with links to numbered results. [AI-generated per source]
- AI Proved What?, The mildest singularity? The ordinary-double-point volume gap (episode 037, October 2026). This AI-assisted video and transcript cover both manuscripts, their scope, earlier special cases, and the all-dimensions argument's stabilizer split. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 037
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-ordinary-double-point-gap-in-every-dimension-September-24-2026,
author = {{OpenAI}},
title = {{The ordinary-double-point gap in every dimension}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-ordinary-double-point-gap-in-every-dimension-September-24-2026/paper.pdf}{OAI:The-ordinary-double-point-gap-in-every-dimension-September-24-2026}},
year = {2026}
}Family 037: The ordinary-double-point volume gap
Proves the ordinary-double-point volume-gap conjecture: every singular complex algebraic klt germ of dimension n ≥ 2, with zero boundary, has normalized volume at most $2(n-1)^n$. Equality holds precisely for an analytic ordinary double point.
Other papers in this family:
- The normalized-volume gap in dimension four · September 24, 2026