In this talk we will introduce and discuss the birational geometry of foliated surfaces. The

purpose of this talk is to show that we can run the minimal model program for foliated surfaces

(with good singularities).

1. We will review the preliminaries for foliated surfaces, e.g. denition of foliations, the canon-

ical divisor of a foliation, reduced singularities and canonical singularities of foliations.

2. We will introduce some formulas for foliated surfaces, e.g. the intersection formulas, the

Camacho-Sad formula.

3. We will show how to run the KF-minimal model program of foliated surfaces.

4. If we still have time we may introduce other related topics, e.g. the classication of foliations

by their (numerical) Kodaira dimension, the failure of nite generation of the canonical ring of

foliations, the cone theorem/partial MMP of high-dimensional foliations, etc.

The key references are the following:

[Bru00] M.Brunella, Birational geometry of foliations, Monografas de Matematica, Instituto

de Matematica Pura e Aplicada(IMPA), Rio de janeiro, 2000.

[Bru02] M.Brunella, Foliations on complex projective surfaces, arXiv:math/0212082, 2002, 31p.

[McQ08] M.McQuillan, Canonical models of foliations. Pure Appl. Math. Q., 4(3, part 2),

2008, pp. 877-1012.