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arXiv · 2608.27341

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 $Z \subseteq X$, it is in general a difficult problem to determine whether $X \setminus Z$ is isomorphic to $\mathbf A^n$. In the case where $X$ is a projective bundle over $\mathbf P^1$ 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.

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Remy van Dobben de Bruyn. 2026-08-27. Divisors in projective bundles over the projective line whose complement is affine space. https://arxiv.org/abs/2608.27341

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