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:Z≥0→Z , then there exists an integer N>0 such that if (X,F) is a canonical or nef model of a foliation of general type with Hilbert polynomial χ(X,OX(mKF))=P(m) for all m∈Z≥0 , then |mKF| defines a birational map for all m≥N On the way, we also prove a Grauert–Riemenschneider-type vanishing theorem for foliated surfaces with canonical singularities.