Effective adjunction theory

Abstract

Here we investigate the property of effectivity for adjoint divisors. Among others, we prove the following results: A projective variety X with at most canonical singularities is uniruled if and only if for each very ample Cartier divisor H on X we have $$H^0(X, m_0K_X+H)=0$$ for some $$m_0=m_0(H)>0$$ . Let X be a projective 4-fold, L an ample divisor and t an integer with $$t \ge 3$$ . If $$K_X+tL$$ is pseudo-effective, then $$H^0(X, K_X+tL) \ne 0$$ .