Explicit bounds on foliated surfaces and the Poincaré problem

We give a solution to the Poincaré 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 $K_{\mathcal{F}}+\tau K_X$.