# Examples of Threefolds with Kodaira Dimension 1 or 2

### Alberto Calabri

Università di Ferrara, Italy### Masaaki Murakami

Universität Bayreuth, Germany### Ezio Stagnaro

Università di Padova, Italy

## Abstract

We construct three nonsingular threefolds $X$, $X_{′}$ and $X_{′′}$ with vanishing irregularities. $X$ has Kodaira dimension $=$ $κ(X)=1$ and its $m$-canonical transformation $φ_{∣mK_{X}∣}$ has the following property: the minimum integer number $m_{0}$, such that the dimension of the image dim $φ_{∣mK_{X}∣}(X)$ $=$ $κ(X)$ $=$ $1$ for $m≥m_{0}$, is given by $m_{0}=32$. $X_{′}$ and $X_{′′}$ have Kodaira dimension $κ(X_{′})=κ(X_{′′})=2$and their $m$-canonical transformations have the properties: dim $φ_{∣mK_{X}∣}(X_{′})$ $=$ $κ(X_{′})$ $=$ $2$ if and only if $m≥12$, dim $φ_{∣mK_{X}∣}(X_{′′})$ $=$ $κ(X_{′′})$ $=$ $2$if and only if $m=9,10$ or $m≥12$.

## Cite this article

Alberto Calabri, Masaaki Murakami, Ezio Stagnaro, Examples of Threefolds with Kodaira Dimension 1 or 2. Rend. Sem. Mat. Univ. Padova 125 (2011), pp. 15–38

DOI 10.4171/RSMUP/125-2