AI 中文总结
本文研究叶状结构的极小模型与翻转，证明秩1叶状结构的$K_F$-MMP输出同构、三维簇余秩1叶状结构的$D$-翻转存在性，同时构造例子揭示秩1叶状结构的病态。
AI 中文摘要
我们研究叶状结构的极小模型与翻转。证明：若$\boldsymbol{\textit{F}}$是正规射影$\boldsymbol{\textit{Q}}$-因子簇上具有典范奇点的秩1叶状结构，且$\boldsymbol{K_F}$是伪有效的，则任意两个$\boldsymbol{K_F}$-MMP的输出同构。对三维簇上的余秩1叶状结构，证明klt情形下$\boldsymbol{D}$-翻转的存在性，且在附加假设下F-dlt情形也成立。相反，我们构造例子表明秩1叶状结构呈现经典MMP中不存在的病态：翻转收缩不一定允许$\boldsymbol{D}$-翻转，且即便在代数空间范畴内，内积且大的典范除子也不一定产生典范模型。
英文摘要
We study minimal models and flops for foliations. We show that if $\mathcal F$ is a rank one foliation with canonical singularities on a normal projective $\mathbb Q$-factorial variety and $K_{\mathcal F}$ is pseudo-effective, then any two outputs of the $K_{\mathcal F}$-MMP are isomorphic. For co-rank one foliations on threefolds, we prove existence results for $D$-flops in the klt setting and, under additional hypotheses, in the F-dlt setting. By contrast, we construct examples showing that rank one foliations display pathologies absent from the classical MMP: flopping contractions need not admit $D$-flops, and nef and big canonical divisors need not give rise to canonical models, even in the category of algebraic spaces.
Comments26 pages
