Characterization of the 4-canonical birationality of algebraic 3-folds, II

TL;DR

This paper investigates the conditions under which the fourth canonical map of nonsingular projective 3-folds of general type with geometric genus 4 is birational, extending previous results for higher genus cases.

Contribution

It provides a characterization of the birationality of the fourth canonical map specifically for 3-folds with geometric genus 4, filling a gap in the existing classification.

Findings

Characterization of birationality for p_g=4 cases

Extension of previous results from p_g≥5 to p_g=4

Conditions under which the fourth canonical map is birational

Abstract

For nonsingular projective 3-folds $X$ (of general type) whose geometric genus $p_g=h^0(X, K_X)$ is $\geq 5$, the birationality of the fourth canonical map $\varphi_{4,X}=\Phi_{|4K_X|}$ was characterized by D.-Q. Zhang and the first author in 2008. This paper aims at characterizing the birationality of $\varphi_{4,X}$ for those $X$ with $p_g=4$.