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Given a projective variety of non-negative Kodaira dimension, it is well-known that the pluricanonical system $|mK_X|$ defines the Iitaka fibration (up to birational equivalence) as long as $m$ is large enough and divisible. It is natural to ask what the precise conditions on $m$ are, so that $|mK_X|$ defines the Iitaka fibration.
The answer to the above problem of effectiveness of Iitaka biratioan is known for curves and surfaces, partially known for treefolds and very little is known for dimension higher. In this talk, I will survey the previous results and techniques of threefolds and explain the new result of Hsin-Ku Chen which shows that $|120K_X|$ defines Iitaka fibrationa for all threefolds with non-negative Kodaira dimension.
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