Explicit bounds on foliated surfaces and the Poincaré problem
Stefania Vassiliadis
TL;DR
The paper tackles the Poincaré problem for foliations by establishing explicit, universal bounds on the degree of general leaves for foliations of general type, using a Minimal Model Program (MMP) for adjoint foliated structures. It introduces and analyzes the $\,\varepsilon$-adjoint framework and proves that the associated pseudo-effective thresholds form a descending chain condition (DCC) set with explicit lower bounds, enabling effective non-vanishing and birationality statements for adjoint divisors $K_{ }+ au K_X$. The results yield explicit birationality bounds and a Viehweg-product-based strategy to handle fibrations, leading to an explicit linear-in-$g$ bound on the degree of leaves in non-isotrivial fibrations on $\mathbb{P}^2$, thereby sharpening the classical Poincaré problem. Overall, the work provides concrete criteria for the effective generation of adjoint linear systems and connects foliated birational geometry with tractable numerical bounds, with broad implications for algebraic foliations and their algebraic solutions.
Abstract
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 $K_{\mathcal{F}} + τK_X$.