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

Abstract

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 $|m(K_V+D)|$. We prove that, for any $m\ge 8$, $|m(K_V+D)|$ gives an ${\mathbb{P}}^1$-fibration form $V$ onto a smooth projective curve.

16 pages, 1 figure