AI 中文总结
该研究探讨等维环态射成为射影空间丛的条件，证明仿射刚性定理并将其应用于长极射线相关的环收缩，得到无需ℚ-因子性的射影空间丛定理。
AI 中文摘要
我们研究等维环态射何时必为射影空间丛。主要结果是仿射刚性定理：若基空间为仿射，环相对典范除子为ℚ-卡蒂埃除子，且其负在每条完全曲线上的次数均大于相对维数，则该态射等变地为平凡射影空间丛。作为应用，针对与长极射线相关的等维环收缩，无需假设ℚ-因子性，即可推导出射影空间丛定理。
英文摘要
We study when an equidimensional toric morphism is forced to be a projective-space bundle. Our main result is an affine rigidity theorem: if the base space is affine, the toric relative canonical divisor is $\mathbb Q$-Cartier, and its negative has degree greater than the relative dimension on every complete curve, then the morphism is equivariantly a trivial projective-space bundle. As an application, we derive a projective-space-bundle theorem for equidimensional toric contractions associated to long extremal rays, without assuming $\mathbb Q$-factoriality.
Comments15 pages
