A Threefold with $p_g=0$ and $P_2=2$

Abstract

We construct a nonsingular threefold $X$ with $q_1=q_2=p_g=0$ and $P_2=2$ whose $m$-canonical transformation ${\phi }_{|m{K}_X|}$ has the following properties

i) ${\phi }_{|m{K}_X|}$ has the generic fiber of dimension $\ge 1$, for $2\le m\le 5$;

ii) it is generically a tranformation 2 : 1, for $6\le m\le 8$ and $m=10$;

iii) it is birational for $m=9$ and $m\ge 11$.

So, we have a gap for $m=10$ in the birationality of ${\phi }_{|m{K}_X|}$.

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