Sarkisov program for algebraically integrable and threefold foliations
Yifei Chen, Jihao Liu, Yanze Wang
TL;DR
The paper extends the classical Sarkisov program to the realm of foliations by developing an MMP framework for adjoint foliated structures. It proves that two Mori fiber spaces arising from an lc algebraically integrable foliation on a $ obreak?_{ ext{``}}$klt variety are linked by a finite sequence of Sarkisov links, with the result extending to dimension $ ext{dim} obreak? ext{} obreak? ext{≤}3$ under mild singularities and incorporating log and adjoint foliated versions. The approach hinges on translating foliated MMP steps into the adjoint foliated setting, establishing terminalizations and finiteness results, and constructing explicit Sarkisov links that terminate. These contributions yield structural insights into birational geometry of foliations and pave the way for understanding foliated Cremona-type groups in higher dimensions. Overall, the work unifies MMP for foliations with Sarkisov-type decompositions and broadens applicability to algebraically integrable adjoint foliated structures.
Abstract
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 aformentioned Sarkisov programs are also established.