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Divisors in projective bundles over the projective line whose complement is affine space

  • Remy van Dobben de Bruyn
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Abstract

Given a smooth projective variety X of dimension n and a closed subscheme Z ⊆ X, it is in general a difficult problem to determine whether X ∖ Z is isomorphic to An. In the case where X is a projective bundle over P1 and the restriction of every irreducible component of Z to every fibre is a hyperplane, we give a complete geometric characterisation in terms of the irreducible components of Z and their intersections. In the appendix, we use this to obtain a coordinate-free explanation for the recent first counterexample to the Jacobian conjecture.

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Cite

BibTeXRIS
BibTeX
@misc{bruyn2026divisors,
  title = {{Divisors in projective bundles over the projective line whose complement is affine space}},
  author = {Remy van Dobben de Bruyn},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.27341},
  eprint = {2608.27341},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.27341},
}
RIS
TY  - UNPB
TI  - Divisors in projective bundles over the projective line whose complement is affine space
AU  - Remy van Dobben de Bruyn
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2608.27341
UR  - https://arxiv.org/abs/2608.27341
AB  - Given a smooth projective variety X of dimension n and a closed subscheme Z ⊆ X, it is in general a difficult problem to determine whether X ∖ Z is isomorphic to A^n. In the case where X is a projective bundle over P^1 and the restriction of every irreducible component of Z to every fibre is a hyperplane, we give a complete geometric characterisation in terms of the irreducible components of Z and their intersections. In the appendix, we use this to obtain a coordinate-free explanation for the recent first counterexample to the Jacobian conjecture.
ER  -