On open algebraic surfaces of general type whose log canonical maps composed of a pencil

Abstract

Let $(S,D)$ be a minimal log pair of general type with $S$ a smooth projective surface and $D$ a simple normal corssing reduced divisor on $S$. We assume that its log canonial linear system $|K_S+D|$ is composed of a penciel, let $f:S\to B$ be the fiberation induced by the linear system $|K_S+D|$ and $F$ be a general fiber of $f$. Let $b$ (resp. $g$) be the genus of the base curve $B$ (resp. general fiber $F$) and $k=D\cdot F$ the intersection number. We show that 1. If $k>0$ and $b\ge 2$ then $2\le g+k\le 3$, when $g+k=3$ we have $b=2$ and $h^1(S,K_S+D)=0$. 2. Suppose $p_a(D)\le 2(l+q(S))+1-h^{1,1}(S)$ where $l$ is the number of irreducible components of $D$, then we have $g\le 5$ for $p_g(S,D)\gg 0$. Moreover if $p_g(S)=0$, then we have $g\le 3$.