Zariski cancellation and affine fibrations over the complex numbers
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
Constructs an integral complex affine fourfold X\not≅\mathbb A^4 with X×\mathbb A^1≅\mathbb A^5, disproving affine-space cancellation over ℂ in dimension four. It also disproves the Dolgachev–Weisfeiler affine-fibration conjecture: smooth surjections X→\mathbb A^1 and \mathbb A^5→\mathbb A^2 have every residue-field fiber isomorphic to affine three-space but are not Zariski-locally trivial.
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MANUSCRIPT 01
An explicit failure of complex affine-space cancellation
We construct an explicit integral complex affine fourfold X with X×\mathbb A^1≅\mathbb A^5 but X\not≅\mathbb A^4. This gives a negative answer to Zariski's affine-space cancellation problem over ℂ in dimension four. The same construction disproves the Stable Coordinate Conjecture in ambient dimension five and yields smooth…
Full source abstract
We construct an explicit integral complex affine fourfold X with X×\mathbb A^1≅\mathbb A^5 but X\not≅\mathbb A^4. This gives a negative answer to Zariski's affine-space cancellation problem over ℂ in dimension four. The same construction disproves the Stable Coordinate Conjecture in ambient dimension five and yields smooth 𝔸^(3)-fibrations over 𝔸^(1) and 𝔸^(2) that are not Zariski-locally trivial. These fibrations disprove the Dolgachev–Weisfeiler affine-fibration conjecture over these bases.
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