On threefolds of general type with \(\chi = 1\) (Q1016515)

In the paper under review, the author proves that if \(V\) is a smooth projective threefold of general type with \(\chi (\mathcal O _V)=1\), then (i) \(P_m(V):=h^0(mK_V)>0\) for all \(m\geq 7\); (ii) \(P_{12}(V)\geq 3\) with three possible exceptions; (iii) \(\phi _m:V\dasharrow \mathbb P H^0(mK_V)\) is birational onto its image for all \(m\geq 46\). It should be noted that (i) (as well as the inequality \(P_m(V)\geq 2\) for all \(m\geq 10\)) was previously proven by different methods in [\textit{D.-K. Shin}, J. Algebra 309, No. 2, 559--568 (2007; Zbl 1116.14032)], and that (iii) is a refinement of results obtained by J. A. Chen and M. Chen and by M. Chen and L. Zhu.