  Pursuing McQuillan's philosophy in proving the Green-Griffiths conjecture for
certain surfaces of general type, we deal with the algebraic degeneracy of
entire curves tangent to holomorphic foliations by curves. Inspired by the
recent work of Paun and Sibony, we study the intersection of Ahlfors current
with the tangent bundle of the foliation, and derive some consequences. In
particular, we introduce the definition of weakly reduced singularities for
foliations by curves, which requires less work than the exact classification
for foliations. Finally we discuss the strategy to prove the Green-Griffiths
conjecture for complex surfaces.
