On minimal 3-folds of general type with maximal pluricanonical section index

TL;DR

This paper investigates minimal projective 3-folds of general type with maximal pluricanonical section index, revealing that their 57th canonical map is stably birational, thus advancing understanding of their birational geometry.

Contribution

It provides a detailed study of 3-folds with maximal pluricanonical section index, establishing the stable birationality of the 57th canonical map for these varieties.

Findings

Maximal pluricanonical section index is either between 1 and 15 or equals 18.

The 57th canonical map of such 3-folds is stably birational.

Enhanced understanding of the birational properties of minimal 3-folds.

Abstract

Let $X$ be a minimal projective 3-fold of general type. The pluricanonical section index $\delta(X)$ is defined to be the minimal integer $m$ so that $P_{m}(X)\geq 2$. According to Chen-Chen, one has either $1\leq \delta(X)\leq 15$ or $\delta(X)=18$. This note aims to intensively study those with maximal such index. A direct corollary is that the $57$th canonical map of every minimal 3-fold of general type is stably birational.