On irregular threefolds and fourfolds with numerically trivial canonical bundle

TL;DR

This paper establishes birationality results for linear systems on irregular threefolds and fourfolds with numerically trivial canonical bundles, extending known results and proving optimal bounds.

Contribution

It provides new birationality criteria for linear systems on irregular threefolds and fourfolds with trivial canonical bundles, using a unified method.

Findings

For 3-folds, |mL+P| is birational for m≥3

For 4-folds, |mL+P| is birational for m≥5

Results are proven to be optimal

Abstract

We prove that for a smooth projective irregular $3$-fold $X$ with $K_X\equiv 0$ and a nef and big divisor $L$ on $X$, $|mL+P|$ gives a birational map for all $m\geq 3$ and all $P\in \text{Pic}^0(X)$. We also use the same method to deal with $4$-folds, and prove that for a smooth projective irregular $4$-fold $X$ with $K_X\equiv 0$ and an ample divisor $L$ on $X$, $|mL+P|$ gives a birational map for all $m\geq 5$ and all $P\in \text{Pic}^0(X)$. These results are also optimal.