Ordinary double point volume gap conjecture

About 3 years old · traced to 2023

Let $n$ be a positive integer, and let an ordinary double point denote the corresponding $n$ -dimensional klt quadratic singularity. Ordinary double point volume gap conjecture. The second largest local volume of an $n$ -dimensional klt singularity is

2(n-1)^n,

and this value is achieved only by the ordinary double point. This refines the expected discreteness of local volumes and would have consequences for K-moduli of cubic hypersurfaces; the source says that even the existence of a volume gap appears nontrivial.

References

Primary source

Ziquan Zhuang, “Stability of klt singularities”, arXiv:2307.10525 (2023).

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RemarkAI-assistedClaimed by OpenAI. For singular boundary-zero complex algebraic klt germs of every dimension n >= 2, the manuscript claims normalized volume at most 2(n-1)^n, with equality if and only if the analytic germ is an ordinary double point. No isolatedness or hypersurface assumption is imposed on the input germ.See full solutionHide full solution

Claimed by OpenAI. For singular boundary-zero complex algebraic klt germs of every dimension n >= 2, the manuscript claims normalized volume at most 2(n-1)^n, with equality if and only if the analytic germ is an ordinary double point. No isolatedness or hypersurface assumption is imposed on the input germ.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-ordinary-double-point-gap-in-every-dimension-September-24-2026/paper.pdf

OpenAI-037-01-The-ordinary-double-point-gap-in-every-dimension.pdf649,493 bytesOpen

RemarkAI-assistedClaimed by OpenAI. For singular boundary-zero complex algebraic klt fourfold germs, the manuscript claims normalized volume at most 162, with equality if and only if the analytic germ is an ordinary double point. This is the dimension-four subcase of the general volume-gap conjecture.See full solutionHide full solution

Claimed by OpenAI. For singular boundary-zero complex algebraic klt fourfold germs, the manuscript claims normalized volume at most 162, with equality if and only if the analytic germ is an ordinary double point. This is the dimension-four subcase of the general volume-gap conjecture.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-normalized-volume-gap-in-dimension-four-September-24-2026/paper.pdf

OpenAI-037-02-The-normalized-volume-gap-in-dimension-four.pdf383,146 bytesOpen