Explicit bounds on foliated surfaces and the Poincar\'e problem

TL;DR

This paper solves the Poincaré Problem by establishing bounds on the degree of leaves in foliations of general type, using birational geometry and the Minimal Model Program to derive effective results.

Contribution

It provides explicit bounds on foliated surface degrees and advances the understanding of the birational geometry of foliations through the study of pseudo-effective thresholds.

Findings

Bound on degree of leaves linear in genus g

Descending chain condition for pseudo-effective thresholds

Effective birationality results for adjoint divisors

Abstract

We give a solution to the Poincar\'e Problem, in the formulation of Cerveau and Lins Neto. We obtain a bound on the degree of general leaves of foliations of general type, which is linear in $g$. To achieve this we study the birational geometry of foliations within the framework of the Minimal Model Program (MMP). Extending the approach of Spicer--Svaldi and Pereira--Svaldi, we study the set of pseudo-effective thresholds of adjoint foliated structures, showing that it satisfies the descending chain condition and it admits an explicit universal lower bound. These results yield effective birationality statements for adjoint divisors of the form $K_{\mathcal{F}} + \tau K_X$.