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 $V\setminus \mathrm{Supp}D=S$. In this paper, we consider the logarithmic multicanonical system $\mathrm{\mid }m(K_V+D)\mathrm{\mid }$. We prove that, for any $m\ge 8$, $\mathrm{\mid }m(K_V+D)\mathrm{\mid }$ gives an ${\mathbb{P}}^1$-fibration form $V$ onto a smooth projective curve.