Effective generation for foliated surfaces: results and applications

We explore the birational structure and invariants of a foliated surface $(X, \mathcal F)$ in terms of the adjoint divisor $K_{\mathcal F}+\epsilon K_X$, $0< \epsilon \ll 1$. We then establish a bound on the automorphism group of an adjoint general type foliated surface $(X, \mathcal F)$, provide a bound on the degree of a general curve invariant by an algebraically integrable foliation on a surface and prove that the set of $\epsilon$-adjoint canonical models of foliations of general type and with fixed volume form a bounded family.