The eventual paracanonical map of a variety of maximal Albanese dimension

TL;DR

This paper investigates the properties of the eventual paracanonical map of varieties with maximal Albanese dimension, revealing its behavior akin to canonical maps on surfaces and establishing birationality for higher multiples.

Contribution

It provides a detailed analysis of the eventual paracanonical map, including its behavior and explicit descriptions in various examples, extending understanding of these maps in algebraic geometry.

Findings

The map behaves like the canonical map for $m=1$ with $ ext{chi}(K_X)>0$.

For $m>1$, the map is birational.

Explicit descriptions are provided in several examples.

Abstract

Let $X$ be a smooth complex projective variety such that the Albanese map of $X$ is generically finite onto its image. Here we study the so-called eventual $m$-paracanonical map of $X$ (when $m=1$ we also assume $\chi(K_X)>0$). We show that for $m=1$ this map behaves in a similar way to the canonical map of a surface of general type, while it is birational for $m>1$. We also describe it explicitly in several examples.