Algebraically integrable foliations link Mori fiber spaces, indicating new connections in geometry.
By applying the theory of the minimal model program for adjoint foliated structures, we establish the Sarkisov program for algebraically integrable foliations on klt varieties: any two Mori fiber spaces of such structure are connected by a sequence of Sarkisov links. Combining with a result of R. Mascharak, we establish the Sarkisov program for foliations in dimension at most $3$ with mild singularities. Log version and adjoint foliated version of the aforementioned Sarkisov programs are also established.
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Chen et al. (2026) studied this question.
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