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European Mathematical Society Publishing House
2016-09-19 17:05:48
Rendiconti del Seminario Matematico della Università di Padova
Rend. Sem. Mat. Univ. Padova
RSMUP
0041-8994
2240-2926
General
10.4171/RSMUP
http://www.ems-ph.org/doi/10.4171/RSMUP
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society (from 2013)
125
2011
0
Examples of Threefolds with Kodaira Dimension 1 or 2
Alberto
Calabri
Università di Ferrara, FERRARA, ITALY
Masaaki
Murakami
Universität Bayreuth, BAYREUTH, GERMANY
Ezio
Stagnaro
Università di Padova, 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 $m \geq 12$.
General
15
38
10.4171/RSMUP/125-2
http://www.ems-ph.org/doi/10.4171/RSMUP/125-2