AI 中文总结
本文研究射影直线上射影丛中补集为仿射空间的除子问题，给出特定情形下的几何刻画，并将其用于解释首个雅可比猜想反例。
AI 中文摘要
给定n维光滑射影簇X及闭子概型Z⊆X，判断X∖Z是否同构于Aⁿ通常是困难问题。当X是P¹上的射影丛，且Z的每个不可约分支在每条纤维上的限制均为超平面时，本文基于Z的不可约分支及其交给出完整几何刻画。附录中，我们用此结果为近期首个雅可比猜想反例提供无坐标解释。
英文摘要
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.
CommentsCorrected the statement and proof of the main theorem. 10 pages. Comments are welcome!
