https://arxiv.org/html/2608.27341
Author not recordedDate not recordedOriginal︎ ↗
Stewards’ notes · 1
- In the 2026 Jacobian-counterexample construction, the complement of the ramification divisor and hyperplane preimage in P¹×P² is isomorphic to affine three-space.
Supports. The preprint proves the isomorphism from a general criterion rather than by coordinates, and adds the converse: the complement is affine three-space exactly when the hyperplane is tangent but not osculating to the small diagonal. The ramification divisor is shown to have bidegree (2,1) and the hyperplane pullback bidegree (1,1), and the tangency condition is shown to make their fibres over the tangency point coincide. The second version corrected the statement and proof of the general theorem after a reader pointed out a mistake; the appendix theorem for the counterexample case is unchanged, and it agrees with the independent coordinate and bundle-tower proofs given elsewhere. Sep 16, 2026
Examinations
None recorded.
Text
Stored, not yet divided into passages.
The complement U = 𝐏 1 × 𝐏 2 ∖ ( R ∪ φ − 1 ( H ) ) U=\mathbf{P}^{1}\times\mathbf{P}^{2}\setminus(R\cup\varphi^{-1}(H)) is isomorphic to 𝐀 3 \mathbf{A}^{3} if and only if H H is tangent but not osculating to the small diagonal.
Supports. The preprint proves the isomorphism from a general criterion rather than by coordinates, and adds the converse: the complement is affine three-space exactly when the hyperplane is tangent but not osculating to the small diagonal. The ramification divisor is shown to have bidegree (2,1) and the hyperplane pullback bidegree (1,1), and the tangency condition is shown to make their fibres over the tangency point coincide. The second version corrected the statement and proof of the general theorem after a reader pointed out a mistake; the appendix theorem for the counterexample case is unchanged, and it agrees with the independent coordinate and bundle-tower proofs given elsewhere.
Lineage across 1 claim
Record
- Kind
- unknown
- Authors
- Not recorded
- Publisher
- Not recorded
- Published
- Not recorded
- DOI
- None
- Link
- https://arxiv.org/html/2608.27341
- Archived copy
- None found
- Stored text
- 49,705 characters, retrieved Sep 16, 2026 · not yet divided
- Versions
- One known
- Copies
- None recorded
- Corrections
- Not yet checked
- Facts checked
- Not yet
History 2
- Sep 16, 2026ReadingSteward of “In the 2026 Jacobian-counterexample constructio…” · supports, read whole
- Sep 16, 2026Entered the graphEntered the graph; copy stored