# Rendiconti del Seminario Matematico della Università di Padova

Volume 125, 2011, pp. 15–38
DOI: 10.4171/RSMUP/125-2

Examples of Threefolds with Kodaira Dimension 1 or 2

Alberto Calabri[1], Masaaki Murakami[2] and Ezio Stagnaro[3]

(1) Dipartimento di Matematica, Università di Ferrara, Via Machiavelli 35, 44100, Ferrara, Italy
(2) Mathematisches Institut, Universität Bayreuth, Universitätsstrasse 30, 95447, Bayreuth, Germany
(3) Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università di Padova, Via Trieste, 63, 35121, Padova, Italy

We construct three nonsingular threefolds $X$, $X'$ and $X''$ with vanishing irregularities. $X$ has Kodaira dimension $=$ $\kappa(X)=1$ and its $m$-canonical transformation $\varphi_{|mK_X|}$ has the following property$\,$: the minimum integer number $m_0$, such that the dimension of the image dim $\varphi_{|mK_X|}(X)$ $=$ $\kappa(X)$ $=$ $1$ for $m \geq m_0$, is given by $m_0=32$. $X'$ and $X''$ have Kodaira dimension $\kappa(X')=\kappa(X'')=2$ and their $m$-canonical transformations have the properties$\,$: dim $\varphi_{|mK_{X'}|}(X')$ $=$ $\kappa(X')$ $=$ $2$ if and only if $m \geq 12$, dim $\varphi_{|mK_{X''}|}(X'')$ $=$ $\kappa(X'')$ $=$ $2$ if and only if $m = 9, 10$ or $m \geq 12$.