A note on non-vanishing and applications

Abstract

Let $X$ be a normal variety over the field of complex numbers with log terminal singularities and the canonical divisor $K_X$ being ${\bf Q}$-Gorenstein. Assume that $L$ is an ample line bundle over $X$ and $ϕ: X\to Y$ is a morphism supported by $K_X+rL$ for some positive rational number $r$. In the present paper we study the evaluation $ϕ^*ϕ_*(L)\to L$ and the locus of points where it is not surjective which we call relative base point locus of $L$. In particular, we prove that, if the dimension of a fiber of $ϕ$ is small with respect to $r$ then the relative base point locus does not meet the fiber. Consequently, in this case, we discuss the structure of the map $ϕ$ for a smooth $X$.