Higher-dimensional foliated Mori theory

Abstract

We develop some foundational results in a higher dimensional foliated Mori theory, and show how these results can be used to prove a structure theorem for the Kleiman-Mori cone of curves in terms of the numerical properties of K F for rank 2 foliations on threefolds. We also make progress toward realizing a minimal model program (MMP) for rank 2 foliations on threefolds.

New directions in the Minimal Model Program

“…In higher dimension, the cone theorem for foliations is known to hold true in two important cases: for rank one foliations in any dimension [8] and for rank two foliations in dimension three [50]. Furthermore, a version of the base point free theorem holds in dimension three by [14,15].…”

Hypersurfaces quasi-invariant by codimension one foliations

“…The second author established in [23] a cone theorem which describes the structure of the Kleiman-Mori cone of curves in terms of numerical properties of the canonical bundle K F of a codimension one foliation F on a projective 3-dimensional variety. We were lead to the definition of quasi-invariant divisors while trying to understand the implications of this result on the geometry/dynamics of the original foliation.…”

“…However, since we only blow up in invariant centres π must be crepant, i.e., K F = π * K F . In particular the strict transform of C is still K F -negative in which case we may apply [23,Corollary 11.2] to conclude that C ⊂ sing(F).…”

“…For simplicity we will consider the case where infinitely many C i meet a curve C tangent to F. The case where infinitely many C i pass through a single point is similar. According to [23,Corollary 6.4], there exists a F-invariant analytic subvariety L containing C which contains the C i . To prove Theorem C it suffices to verify that L is algebraic and rational.…”

“…The extremal rays detected by Theorem 3 are of three different types according to the dimension ofloc(R) = {x ∈ X : x ∈ C such that [C] ∈ R} ,the locus of points belonging to a curve C with class spanning the extremal ray R. In the terminology of[23, Definition 23], a K F -negative extremal ray can be of one of the following types.1. Fiber type when dim loc(R) = 3.…”