On birational geometry of minimal threefolds with numerically trivial canonical divisors

TL;DR

This paper studies the birational geometry of minimal threefolds with numerically trivial canonical divisors, establishing conditions under which certain linear systems induce birational maps.

Contribution

It proves that for such threefolds, the linear systems |mL| and |K_X + mL| are birational for all m ≥ 17, even under weaker assumptions on L.

Findings

|mL| and |K_X + mL| are birational for m ≥ 17

Results hold under weaker assumptions on L being big with no stable base components

Advances understanding of birational maps on minimal threefolds with trivial canonical class

Abstract

For a minimal $3$-fold $X$ with $K_X\equiv 0$ and a nef and big Weil divisor $L$ on $X$, we investigate the birational geometry inspired by $L$. We prove that $|mL|$ and $|K_X+mL|$ give birational maps for all $m\geq 17$. The result remains true under weaker assumption that $L$ is big and has no stable base components.