Effective non-vanishing conjectures for projective threefolds

Abstract

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 pseudo-effective, 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.