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 transformationφ|m__KX| has the following properties
i) φ|m__KX| has the generic fiber of dimension ≥ 1, for 2 ≤ m ≤ 5;
ii) it is generically a tranformation 2 : 1,for 6 ≤ m ≤ 8 and m = 10;
iii) it is birational form = 9 and m ≥ 11.