On projective 3-folds of general type with small positive geometric genus

TL;DR

This paper investigates pluricanonical maps of minimal projective 3-folds of general type with small positive geometric genus, establishing birationality results for specific pluricanonical maps and identifying exceptions.

Contribution

It extends classical results by providing a finite classification of exceptions and proving birationality of certain pluricanonical maps for low geometric genus 3-folds.

Findings

Birationality of 16, 6, and 5 pluricanonical maps for genus 1, 2, and 3 respectively.

Finite list of weighted baskets of exceptions.

Extension of classical results on pluricanonical maps to new cases.

Abstract

In this paper we study pluricanonical maps of minimal projective 3-folds of general type with geometric genus $1$, $2$ and $3$. We go in the direction pioneered by Enriques and Bombieri, and other authors, pinning down, for low projective genus, a finite list of exceptions to the birationality of some pluricanonical map. In particular, apart from a finite list of weighted baskets, we prove the birationality of $\varphi_{16}$, $\varphi_{6}$ and $\varphi_{5}$ respectively.