On birational boundedness of foliated surfaces

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