Effective generation for foliated surfaces: results and applications

Abstract

We prove some results on the birational structure and invariants of a foliated surface (X, F) in terms of the adjoint divisor KF + ǫKX , 0 < ǫ ≪ 1. Several applications of these ideas are considered as well. Contents 1. Introduction 1 2. Preliminaries 4 3. Adjoint MMP 14 4. A bound on the pseudo-effective threshold in the big case 17 5. Applications 23 6. Boundedness of ample models 24 References 30

Analytic varieties invariant by foliations and Pfaff systems

“…Such result is also related with the problem of bounding the order of the automorphism group of foliations of general type, see [35,37,85,103]. In the recent paper [67] the authors prove a bound for other birational invariant of a foliation, called the slope, which is related to the Poincaré problem.…”