Counterexamples to the MMP for 1-foliations in positive characteristic

We show that many statements of the Minimal Model Program, including the cone theorem, the base point free theorem and the existence of Mori fibre spaces, fail for 1-foliated surface pairs \((X,\mathcal {F})\) $(X,\mathcal{F})$ with canonical singularities in characteristic \(p>0\) $p>0$ .