AI 中文总结
本文证明特征零域上光滑射影变体的单有理性、有理连通性与有理链连通性等价，通过引入MU(X)并结合MRC纤维丛与归纳论证完成证明。
AI 中文摘要
本文证明了在特征零域k上的光滑射影变体中，单有理性、有理连通性与有理链连通性三者等价。我们的方法利用MRC纤维丛，证明若X为光滑射影变体，则存在变体MU(X)，连同有理映射π: X ⇢ MU(X)和λ: MU(X) ⇢ MRC(X)，满足：i) 若ν: X ⇢ MRC(X)，则λ∘π在合适的定义域上等于ν；ii) π的非常一般纤维是单有理的；iii) λ的非常一般纤维是有理连通但非单有理的。随后我们运用归纳论证证明MU(X)与MRC(X)双有理等价。
英文摘要
In this paper we prove that unirationality, rational connectedness and rational chain connectedness coincide for smooth projective varieties over a field $ k $ of characteristic zero. Our approach uses the MRC fibration to show that if $ X $ is a smooth projective variety, then there exists a variety $ \operatorname{MU}(X) $, together with rational maps $ π: X \dashrightarrow \operatorname{MU}(X) $ and $ λ: \operatorname{MU}(X) \dashrightarrow \operatorname{MRC}(X) $, such that i) if $ ν: X \dashrightarrow \operatorname{MRC}(X) $, then $ λ\circ π= ν$ on the appropriate domains; ii) the very general fibres of $ π$ are unirational; iii) the very general fibres of $ λ$ are rationally connected but not unirational. We then apply an induction argument to show that $ \operatorname{MU}(X) $ is birationally equivalent to $ \operatorname{MRC}(X) $.
