Projective normality of toric 3-folds with non-big adjoint hyperplane sections, II

Let (X, A) be a nonsingular polarized toric 3-fold. We show that if the adjoint bundle of A has no global sections, then all ample line bundles on X are normally generated. Even if the adjoint bundle of A is effective, if it is not big, then it is also shown the normal generation. Especially, we show that all ample line bundles on a nonsingular toric Fano 3-fold are normally generated.