Examples of threefolds with Kodaira dimension 1 or 2
Let \(X\) be an \(n\)-dimensional smooth complex projective variety of general type. Then it is a recent important result that there exists a positive constant \(r_n\) which depends only on \(n\) such that, for every \(r\geq r_n\), the pluri-canonical map \(\phi_r: X\dashrightarrow\mathbb{P}H^0(X,rK_X)\) is birational, see [\textit{C. D. Hacon} and \textit{J. McKernan}, Invent. Math. 166, No. 1, 1--25 (2006; Zbl 1121.14011)], [\textit{S. Takayama}, Invent. Math. 165, No. 3, 551--587 (2006; Zbl 1108.14031)] and [\textit{H. Tsuji}, Osaka J. Math. 44, No. 3, 723--764 (2007; Zbl 1186.14043)]. The paper under review deals with a similar question for varieties which are not of general type: the question is to establish the minimum number \(\mu_n\) such that the image of \(X\) under the pluri-canonical map \(\phi_r\) has the dimension equal to \(\kappa(X)\) for every \(r\geq\mu_n\). The authors construct three threefolds and compute the corresponding constants \(\mu_3\) for them. The first threefold \(X\) has \(\kappa(X)=1\), and implies that \(\mu_3\geq32\) for threefolds with Kodaira dimension \(1\). The other two threefolds have Kodaira dimension \(2\), and the computation yields that the corresponding constant \(\mu_3\) is at least \(12\).
scientific article; zbMATH DE number 5961386
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