Foliated Minimal Models and Flops

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.