OpenAI · 059-01 · 2026-09-24 · Algebraic and complex geometry
Ambiently homeomorphic isolated hypersurfaces of multiplicities two and three
preprints/Ambiently-homeomorphic-isolated-hypersurfaces-of-multiplicities-two-and-three-September-24-2026
Family 059. Counterexamples to Zariski's multiplicity conjecture.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Disproves Zariski's multiplicity conjecture by constructing reduced holomorphic hypersurface germs that are ambiently homeomorphic but have different multiplicities. The examples include hypersurfaces in with isolated critical points and multiplicities four and five.
We give a negative answer to the embedded Zariski multiplicity conjecture. We construct two reduced hypersurface germs that are ambiently homeomorphic but have multiplicities two and three. Both have isolated singularities and lie in a common complex affine space of dimension divisible by eight. Their defining function germs are also topologically right equivalent, and the same examples answer Arnold's corank problem negatively for ambient topological equivalence.
Preprint source and Lean scope
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