The eventual map for irregular varieties

This thesis aims to prove the existence of the eventual map associated with a line bundle on a projective irregular variety. In addition we want to study its relations with some classical invariants, in particular in the case of surfaces. The eventual map is a new way to associate a morphism to a line bundle L on a normal irregular variety X. The idea is to use the existence of a non trivial morphism a: X → A into an abelian variety A in order to “perturb” the line bundle and see what features “stay the same”. After the proof of existence, we will study the m-paracanonical map for surfaces φm, i.e. the eventual map associated to the m-canonical line bundle ω_S^⊗m. We are going to study what happen by removing the finiteness assumption of a. If φm is generically finite, we will bound its degree, and if it is composed with a pencil we will find a bound for genus of the pencil. Jiang in [Jia21, Lemma 2.10] proves that the genus of the pencil is between 2 and 5. We will improve this result by establishing what are the condition to be genus 4 and 5. We will conclude this work whit some examples, in order to show how much the eventual map is concrete.