On a conjecture of Hacon and McKernan in dimension three

We prove that there exists a universal constant $r_3$ such that if $X$ is a smooth projective threefold over $\mathbb{C}$ with non-negative Kodaira dimension, then the linear system $\mathrm{\mid }r{K}_X\mathrm{\mid }$ admits a fibration that is birational to the Iitaka fibration as soon as $r\ge r_3$ and sufficiently divisible. This gives an affirmative answer to a conjecture of Hacon and McKernan in the case of threefolds. Viehweg and Zhang have posted a stronger result along these lines using different methods.