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

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

Let $L$ be an ample line bundle on a nonsingular toric 3-fold. We show that if the adjoint bundle of $L$ has no global sections, then $L$ is normally generated. Even if the adjoint bundle is effective, it is shown that $L$ is normally generated if it is not big.