On birational boundedness of foliated surfaces

In this paper we prove a result on the effective generation of pluri-canonical linear systems on foliated surfaces of general type. Fix a function $P:{\mathbb{Z}}_{\ge 0}\to \mathbb{Z}$, then there exists an integer $N_1>0$ such that if $(X,\mathcal{F})$ is a canonical or nef model of a foliation of general type with Hilbert polynomial $\chi (X, mK_{\mathcal F})=P(m)$forall$m\in \mathbb Z_{\geq 0}$,then$|mK_{\mathcal F}|$ defines a birational map for all $m\ge N_1$. We also prove a Grauert-Riemannschneider type vanishing theorem for foliated surfaces with canonical singularities.