On an open problem of characterizing the birationality of 4K

TL;DR

This paper characterizes when the 4-canonical map of a minimal projective 3-fold of general type is non-birational, linking it to the existence of a specific pencil of surfaces, and clarifies the genus condition for this property.

Contribution

It proves a precise criterion for the birationality of the 4-canonical map in terms of fibrations by (1,2)-surfaces for 3-folds with genus at least 5.

Findings

4-canonical map non-birational iff the 3-fold is fibered by (1,2)-surfaces for genus ≥ 5.

Counterexamples show the criterion does not hold for genus ≤ 4.

Provides a complete answer to an open problem posed by Chen and Zhang in 2008.

Abstract

We answer an open problem raised by Chen and Zhang in 2008 and prove that, for any minimal projective 3-fold $X$ of general type with the geometric genus $\geq 5$, $X$ is birationally fibred by a pencil of $(1,2)$-surfaces (i.e. $c_1^2=1$, $p_g=2$) if and only if the $4$-canonical map $\varphi_{4,X}$ is non-birational. The statement does not hold for those with the geometric genus $\leq 4$ according to our examples.