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Non-flat toroidal compactifications of the universal abelian variety in genus five and above

Primarily AI-generated textHuman understanding: some partsmath.AG — Algebraic Geometry

Contributed by David Holmes ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

We prove that two mixed Delaunay--Voronoi models of the universal abelian variety are not flat in genus five: the classical model over the second-Voronoi compactification and the model over the finite cover that records a symmetric theta divisor. Both counterexamples arise from the same monoid-theoretic obstruction: two inequivalent minimal elements in one fibre-degree slice acquire a common positive translate. The ordinary Delaunay cells and cones are certified by exact all-lattice-point computations; after this geometric input, the obstruction is verified by the integer calculations displayed below. Combining the genus-five example in the fixed-theta model with a theorem of Nakamura--Sugawara gives the sharp threshold: the morphism is flat exactly in genera at most four.

Provenance statement

A first search for these examples was made in June 2026 with Opus 4.8 (via Claude Code), which required quite a lot of help to correctly understand the mathematics, and to make the code more efficient. Code was reviewed by GPT 5.5 (via Codex). Some searches were made, but no counterexample found. In September GPT 5.6 was used to review the code and make the search more efficient. It found the counterexamples given here within a day, and wrote a first (extremely) rough version of this paper. Fable 5.1, Opus 5, and Opus 5.5 were used to improve the writeup and check the literature for prior work, with repeated refereeing by GPT 5.6 and 6. As part of the verification, the argument was re-written in Lamport's formal proof notation. I believe that I understand the mathematics well, in terms of the construction of the toroidal compactifications, the flatness criteria, etc. But the computer search was carried out largely independently by AI, and I have not checked the code to any meaningful degree. I have checked that the writeup seems broadly correct, but have not verified every line myself.

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Version history

  1. v1Submitted by David HolmesInitial depositCurrentOct 19, 2026