A note on the linear systems on the projective bundles over Abelian varieties

TL;DR

This paper generalizes classical results on linear systems of ample line bundles over Abelian varieties, providing new proofs using the CGG theory, and extends understanding of the base point freeness and very ampleness of these systems.

Contribution

It introduces a generalized framework for linear systems on projective bundles over Abelian varieties, utilizing the CGG theory for new proofs and insights.

Findings

|2L| is base point free for ample L on Abelian varieties

3L is very ample on Abelian varieties

New proof techniques using CGG theory

Abstract

It is well known that for an ample 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. In this paper we generalized this classical result and give a new proof using the theory CGG developed by Pareschi and Popa.