Effective non-vanishing conjectures for projective threefolds

Abstract

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 + 2 A is nef, the adjoint bundle K X + 2 A has global sections.