Examples of Threefolds with Kodaira Dimension 1 or 2

Abstract

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