The map defined by a non-very ample line bundle on an irregular variety

TL;DR

This paper investigates the properties of maps defined by non-very ample line bundles on irregular varieties, extending classical results from Abelian varieties to projective bundles and analyzing the bicanonical map of certain irregular varieties.

Contribution

It generalizes classical results about line bundles on Abelian varieties to projective bundles and explores the bicanonical map of irregular primitive varieties of general type.

Findings

The map defined by |2L| on Abelian varieties is well understood.

Generalization of classical results to projective bundles over Abelian varieties.

Relation between the bicanonical map and divisor reducibility on irregular varieties.

Abstract

In this paper, we studied the map defined by a non-very ample line bundle on some special irregular varieties. As to this topic, it is well known that for a line bundle $L$ on an Abelian variety $A$, the linear system $|2L|$ is base point free, and 3L is very ample, moreover the map defined by the linear system $|2L|$ is well understood (cf. Theorem \ref{oldth}). First, we generalized this classical result to projective bundles over Abelian varieties (cf. Theorem \ref{key}). Then we studied the bicanonical map of an irregular primitive variety $X$ of general type with $dim(X) = q(X)$, in fact we got a relation between the map and the reducibility of a divisor.