On an open problem of characterizing the birationality of $4K$

We answer an open problem raised by Chen–Zhang in 2008 and prove that, for any minimal projective $3$-fold $X$ of general type with the geometric genus $p_g (X) \geq 5$, the $4$-canonical map $\varphi_{4,X}$ is non-birational if and only if $X$ is birationally equivalent to a fibration, onto a curve, of which the general fiber is a minimal surface of general type with $(c^2_1, p_g) = (1, 2)$. The statement does not hold for those with the geometric genus $p_g (X) \leq 4$ according to our examples.

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The first author was supported by National Natural Science Foundation of China (#12071078, #11571076, #11731004, #11421061) and Program of Shanghai Subject Chief Scientist (#16XD1400400).

The second author is supported by National Researcher Program of National Research Foundation of Korea (Grant No. 2010- 0020413) and the Shanghai Pujiang Program Grant No. 21PJ1405200.

Received 17 June 2018

Accepted 29 April 2019

Published 17 May 2022