For hyper-Kähler fourfolds of K3[2] type with Picard rank two and two divisorial contractions, the pair of conic-bundle types must be one of seven combinations; for Fano varieties of cubic fourfolds, only four combinations occur.
MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations , Invent.\ Math.\ 198 (2014) 505--590
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The two extremal rays of some Hyper-K\"ahler fourfolds
For hyper-Kähler fourfolds of K3[2] type with Picard rank two and two divisorial contractions, the pair of conic-bundle types must be one of seven combinations; for Fano varieties of cubic fourfolds, only four combinations occur.