Characterization of the 4-canonical birationality of algebraic threefolds

Abstract

In this article we present a 3-dimensional analogue of a well-known theorem of Bombieri (Inst Hautes Etudes Sci Publ Math 42:171-219, 1973) which characterizes the bi-canonical birationality of surfaces of general type. Let X be a projective minimal threefold of general type with ℚ-factorial terminal singularities and the geometric genus p g X 5. We show that the 4-canonical map 4 is not birational onto its image if and only if X is birationally fibred by a family C of irreducible curves of geometric genus 2 with KXC 0 =1 where C 0 is a general irreducible member in C. © 2007 Springer-Verlag.