Characterization of the 4-canonical birationality of algebraic threefolds, II

Abstract

For nonsingular projective threefolds X (of general type) whose geometric genus p g = h 0 (X, K X ) is ≥5, the birationality of the fourth canonical map ϕ 4,X = |4K X | was characterized by D.-Q. Zhang and the first author in 2008. This paper aims at characterizing the birationality of ϕ 4,X for those X with p g = 4.

On an open problem of characterizing the birationality of 4K

“…However the difficulty is how to prove the non-existence of a pencil of (1, 2)-surfaces on a 3-fold. For the case of p g = 4, the reader may refer to [CZ16] for a complete characterization of the birationality of ϕ 4 .…”

On projective threefolds of general type with small positive geometric genus

“…The number deg c (F) = (K n−s W Question A. (see [13,Problem 3.20]) Let X be a minimal projective 3-fold of general type.…”