Abstract
We prove existence of flips, special termination, the base point free theorem and, in the case of log general type, the existence of minimal models for F-dlt foliated pairs of co-rank one on a {mathbb {Q}}-factorial projective threefold. As applications, we show the existence of F-dlt modifications and F-terminalisations for foliated pairs and we show that foliations with canonical or F-dlt singularities admit non-dicritical singularities. Finally, we show abundance in the case of numerically trivial foliated pairs.
Highlights
The Minimal Model Program predicts that a complex projective manifold is either uniruled or it admits a minimal model, i.e. it is birational to a projective variety with nef canonical divisor
Assuming that X is a normal complex projective variety and F is a foliation with mild singularities, it is conjectured that either F is uniruled, i.e. X is covered by rational curves which are tangent to F, or F admits a minimal model, i.e. X is birational to a projective variety Y such that the transformed foliation on Y
We are unable to show termination in complete generality, but we are able to show a weaker version of termination, i.e., termination of flips with scaling, which suffices to show that minimal models exist in several cases of interest: Theorem 1.2 (= Theorem 10.3 + Theorem 11.3) Let F be a co-rank one foliation on a Q-factorial projective threefold X
Summary
The (classical) Minimal Model Program predicts that a complex projective manifold is either uniruled or it admits a minimal model, i.e. it is birational to a (possibly singular) projective variety with nef canonical divisor. This is still an open problem, many important cases of the program have been carried out successfully, e.g. in the case of varieties of dimension at most three and for varieties of general type. Many of the main goals of the program were carried out successfully in the case of rank one foliations (cf [30,31]) and, in any rank, it is expected to follow the main steps of Mori’s program. The goal of this paper is to show the existence of flips (cf. Sect. 2.6) for foliations of co-rank one on a complex projective threefold and present several applications, under some natural assumptions on the singularities
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