Boundedness of fibers for pluricanonical maps of varieties of general type

TL;DR

This paper establishes bounds on the fibers of pluricanonical maps for varieties of general type across dimensions, showing they are birationally bounded for certain r-values.

Contribution

It proves that pluricanonical maps have birationally bounded fibers for threefolds and fourfolds, extending these results to higher dimensions with natural constants.

Findings

Birational boundedness of fibers for threefolds when r ≥ 2

Birational boundedness of fibers for fourfolds when r ≥ 4

Extension of results to higher dimensions

Abstract

We prove that the r-th pluricanonical maps of threefolds of general type have birationally bounded fibers if $r\geqslant 2$. Similarly, we prove that the r-th pluricanonical maps of fourfolds of general type have birationally bounded fibers if $r\geqslant 4$. We extend these results to higher dimensions in terms of constants arising naturally from the birational geometry of varieties of general type.