On canonically derived families of surfaces of general type over curves

Abstract

Suppose X is a smooth projective 3-fold of general type and |mK_X| is composed of a pencil of surfaces with m>1. This pencil naturally induces a fibration f:X->C onto a smooth curve C after the Stein-factorization, which is the main objects of this article. Based on Koll'ar's earlier works, we improve on it and try to understand the family in terms of discrete birational invariants of the total space as well as those of the general fiber and the base curve. The aim of this note is to build a little bit basic facts.