Let \({\mathcal L}\) be an ample and spanned line bundle on a normal connected Gorenstein projective n-fold Z. Assume that \(n\geq 3\), that cod(Sing(Z))\(\geq 3\), and that Irr(Z), the locus of non rational singularities, is finite. It is shown that \(h^ 0((K_ Z\otimes {\mathcal L}^{n-2})^ N)=0\) for all \(N>0\) if and only if \(h^ 0(K_ Z\otimes {\mathcal L}^{n-2})=0\). Previous results of the author [in Computer analysis and algebraic geometry Proc. Conf., Göttingen 1985, Lect. Notes Math. 1194, 175-213 (1986; Zbl 0601.14029)] classified the pairs (Z,\({\mathcal L})\) where \(h^ 0((K_ Z\otimes {\mathcal L}^{n-2})^ N)=0\) for all \(N>0\). Various corollaries about the invariants of surface sections of n-folds are obtained.