  Let $X$ be a smooth projective $n$-fold such that $q(X)=0$ and $L$ a globally
generated, big line bundle on $X$ such that $h^0(K_X+(n-2)L) >0$. We give
necessary and sufficient conditions for the adjoint systems $|K_X+kL|$ to be
birational for $k \geq n-1$. In particular, for Calabi-Yau $n$-folds we
generalize and prove parts of a conjecture of Gallego and Purnaprajna.
