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Effectiveness of the log Iitaka fibration for 3-folds and 4-folds. (English) Zbl 1184.14023

The geometry of pluricanonical maps is an important tool to study algebraic varieties. In recent years many progress have been achieved in the study of pluricanonical maps of varieties of general type [see C. D. Hacon, J. McKernan, Invent. Math. 166, No. 1, 1–25 (2006; Zbl 1121.14011); S. Takayama, Invent. Math. 165, No. 3, 551–587 (2006; Zbl 1108.14031); H. Tsuji, Pluricanonical systems of projective varieties of general type, arXiv:math.AG/09909021]. The paper under review is addressed to non uniruled varieties not of general type. In this class the pluricanonical sections define, up to birationality, a unique fiber space called the Iitaka fibration. It is an interesting and open question weather there exists a universal constant, say \(r(n)\), depending only on the dimension of the variety \(X\), such that the \(r(n)^{th}\) pluricanonical map is birational to the Iitaka fibration, see also [G. Pacienza, Math. Res. Lett. 16, No. 4, 663–681 (2009; Zbl 1184.14022)]. The precise contribution of the paper is too technical to be contained in a review. The main results can be expressed as follows. The authors prove the existence of a computable \(r(n)\) for \(n\leq 4\) and under the assumption that the Kodaira dimension of the variety is bounded by \(2\). The result is obtained by a skillful analysis of the surface case.

MSC:

14E05 Rational and birational maps
14J35 \(4\)-folds
14J30 \(3\)-folds
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