Chen, Y., Liu, J., & Wang, Y. (2024). Flop between algebraically integrable foliations on potentially klt varieties. arXiv preprint arXiv:2410.05764v1.
This paper investigates the relationship between different minimal models of algebraically integrable foliations on potentially klt varieties. The main research question is whether these minimal models are connected by a sequence of flops, similar to the established results for varieties.
The authors employ techniques from algebraic geometry, specifically the minimal model program (MMP) for foliations. They adapt Kawamata's proof for the flop connection between minimal models of varieties to the context of algebraically integrable foliations. The proof involves constructing specific MMPs, analyzing discrepancies of divisors, and utilizing properties of generalized foliated quadruples.
The paper's main finding is a proof that any two minimal models of an lc algebraically integrable foliated triple on potentially klt varieties are connected by a sequence of flops. This result holds for both Q-factorial and non-Q-factorial varieties.
The authors conclude that the connection between minimal models via flops, previously established for varieties, extends to algebraically integrable foliations on potentially klt varieties. This finding contributes to the understanding of the geometry of foliations and the broader minimal model program.
This research advances the minimal model program for foliations by establishing a key connection between different minimal models. It provides a deeper understanding of the birational geometry of foliated varieties and opens avenues for further research in this area.
The paper primarily focuses on algebraically integrable foliations. Future research could explore whether similar flop connections exist for non-algebraically integrable foliations. Additionally, extending these results to higher-dimensional varieties and more general settings within the minimal model program presents further research opportunities.
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by Yifei Chen, ... at arxiv.org 10-10-2024
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