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

TL;DR

This paper investigates the structure of minimal log pairs of general type with log canonical maps composed of a pencil, establishing bounds on the genus of fibers and base curves based on intersection properties and invariants.

Contribution

It provides new bounds on the genus of fibers and base curves for such surfaces, linking geometric properties with algebraic invariants in the context of log pairs.

Findings

If $k>0$ and $b extgreater=2$, then $2 extless= g+k extless= 3$.

When $g+k=3$, the base curve has genus 2 and $h^1(S,K_S+D)=0$.

For certain conditions on $p_a(D)$, the genus $g$ is bounded above by 5, or 3 if $p_g(S)=0$.

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 $f\colon S\to B$ 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 $b\geq 2$ then $2\leq g+k \leq 3$, when $g+k=3$ we have $b=2$ and $h^1(S,K_S+D)=0$. 2. Suppose $p_a(D)\leq 2(l+q(S))+1-h^{1,1}(S)$ where $l$ is the number of irreducible components of $D$, then we have $g\leq 5$ for $p_g(S,D)\gg 0$. Moreover if $p_g(S)=0$, then we have $g\leq 3$.