fibration for pair GMV3RTID

fibration for pair

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Theorem 1.1 (Lengths of extremal rational curves for toric foliated pairs). Let X be a projective (not necessarily Q-factorial) toric variety and let (F , ∆) be a log canonical toric foliated pair on X with rankF = r. Then l(F,∆)(R) := min [C]∈R{−(KF + ∆) · C} ≤ r + 1 holds for every extremal ray R of the Kleiman–Mori cone NE(X) = NE(X). Moreover, if l(F,∆)(R) > r holds for some extremal ray R of NE(X), then the contraction morphism φR : X → Y associated to R is a Pr-bundle over Y . In this case, F = TX/Y holds, where TX/Y is the relative tangent sheaf of φR : X → Y , and the sum of the coefficients of ∆ is less than one. In particular, the foliation F is locally free. (pdf) (Fujino 和 Sato, 2024, p. 1)