Adjoint divisors on algebraic curves

Let C be a numerically connected curve lying on a smooth algebraic surface. We show that if is an ample invertible sheaf satisfying some technical numerical hypotheses then is normally generated. As a corollary we show that the sheaf ω_C^⊗2 on a numerically connected curve C of arithmetic genus p_a?3 is normally generated if ω_C is ample and does not exist a subcurve B⊂C such that p_a(B)=1=B(C−B).