A note on non-vanishing and applications

TL;DR

This paper investigates the properties of the relative base point locus of an ample line bundle over a normal variety with log terminal singularities, establishing conditions under which it avoids certain fibers and analyzing the structure of the associated morphism.

Contribution

It introduces the concept of the relative base point locus and proves its non-intersection with fibers of small dimension relative to a parameter r, providing new insights into the structure of morphisms on such varieties.

Findings

The relative base point locus does not meet fibers with small dimension relative to r.

Conditions under which the morphism's structure can be described explicitly.

Insights into the behavior of line bundles on varieties with log terminal singularities.

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 $\phi: X\to Y$ is a morphism supported by $K_X+rL$ for some positive rational number $r$. In the present paper we study the evaluation $\phi^*\phi_*(L)\to L$ 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 $\phi$ 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 $\phi$ for a smooth $X$.