Counterexamples to the MMP for 1-foliations in positive characteristic (Q6592696)

The minimal model program (for varieties) is expected to exist and have nice properties both in characteristic zero and in characteristic \(p > 0\) (in suitable formulations). If we consider foliations instead (that is, subsheaves \(\mathcal F\) of the tangent sheaf closed under Lie bracket), we can still associate to \(\mathcal F\) a canonical divisor \(K_{\mathcal F}\) and therefore formulate conjectures analogous to the MMP. This is known as the MMP for foliations.\N\NA lot is already known about the MMP for foliations in characteristic zero. In the present paper, the author shows that even the most basic statements, such as the Cone Theorem in dimension two, fail for foliations if the characteristic is positive. He also speculates that the correct framework for these questions might be that of \(\infty\)-foliations, but that is not the subject of this paper.