On a computation of plurigenus of a canonical threefold (Q2787664)

Let \(X\) be a complex projective \(3\)-fold with an ample canonical divisor \(K_X\) which has at worst canonical singularities. Let \(p_n(X)= h^0(X, \mathcal{O}_X(nK_X))\) be the \(n\)-th plurigenus of \(X\). According to a result of \textit{J. Kollár} [Ann. of Math. 123, No. 1, 11--42 (1986; Zbl 0598.14015)] if \(p_n(X)\geq2\) for some \(n\) then the (\(11n+5\))-pluricanonical map of \(X\) is birational. The paper under review shows that \(p_n(X)>0\) for some \(n\in \{6, 8, 10\}\), and if \(p_2(X)>0\) or \(p_3(X)>0\) then \(p_{12}(X)\geq2\) and \(p_n(X)\geq2\) for \(n\geq14\) with a possible exceptional case.