Canonical Dimension of an Elliptic or Quasielliptic Fibration

Abstract

Let f: X → B be an elliptic or quasielliptic fibration. Theorem 7.15 expresses the dualizing sheaf ω _x of X in the form

$${\omega _x} = f*({L^{ - 1}} \otimes {\omega _B}) \otimes {{\text{O}}_x}(\sum\limits_{i = 1}^r {{a_i}{p_i}} )0{a_i} < {m_i}$$