  Let X be a smooth projective threefold, and let A be an ample line bundle
such that $K_X+A$ is nef. We show that if $K_X$ or $-K_X$ is pseudoeffective,
the adjoint bundle $K_X+A$ has global sections. We also give a very short proof
of the Beltrametti-Sommese conjecture in dimension three, recently proven by
Fukuma: if A is an ample line bundle such that $K_X+2A$ is nef, the adjoint
bundle $K_X+2A$ has global sections.
