Logarithmic multicanonical systems of smooth affine surfaces of logarithmic Kodaira dimension one

Let S be a smooth affine surface of logarithmic Kodaira dimension one and let $(V,D)$ be a pair of a smooth projective surface V and a simple normal crossing divisor D on V such that V Supp D = S. In this paper, we consider the logarithmic multicanonical system |m(KV + D)|. We prove that, for any m ≥ 8, |m(KV+D)| gives an P¹-fibration form V onto a smooth projective curve.