Result 037, Algebraic and complex geometry

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)n2(n-1)^n. Equality holds precisely for an analytic ordinary double point.

Proof

The bigger picture

Why it matters

A singularity is a point where a geometric space fails to be smooth; a germ describes the space near that point. This result identifies the singular germs with the largest possible value of a local geometric invariant.

What changes?

The manuscript reports that every singular complex algebraic germ of dimension n at least two, within the mild-singularity class called klt and with zero boundary, has normalized volume at most 2 times (n minus 1) to the nth power. Normalized volume is a numerical invariant of the local singularity; zero boundary means no additional divisor data enter the pair. Equality holds exactly for analytic ordinary double points. In dimension four, the bound is 162.

What does that help mathematicians do?

An ordinary double point has a local analytic description by a nondegenerate quadratic equation. The claimed equality characterization therefore turns an extremal numerical value into a precise description of local geometry. Within the stated class, a singularity attaining the bound must have that form, while every other singularity has strictly smaller normalized volume. This does not assert a uniform further gap below the maximum.

Are there practical applications?

The immediate value is foundational: the result supplies a sharp benchmark for comparing local algebraic singularities in every dimension at least two. Researchers computing normalized volumes can use the bound to check possible values and the equality case to identify extremal geometry. The supplied abstracts do not describe a practical deployment or computational algorithm.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

The ordinary-double-point gap in every dimension

September 24, 2026 57 pages

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)n2(n-1)^n, with equality precisely at an analytic ordinary double point.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

The normalized-volume gap in dimension four

September 24, 2026 20 pages

We prove that every singular complex algebraic klt fourfold germ with zero boundary has normalized volume at most 162, with equality precisely for an analytic ordinary double point. This resolves the ordinary-double-point volume-gap conjecture in dimension four.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:The-normalized-volume-gap-in-dimension-four-September-24-2026,
  author = {{OpenAI}},
  title = {{The normalized-volume gap in dimension four}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-normalized-volume-gap-in-dimension-four-September-24-2026/paper.pdf}{OAI:The-normalized-volume-gap-in-dimension-four-September-24-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.