On the bicanonical maps of primitive varieties with $q(X) = dim(X)$: the degree and the Euler number

TL;DR

This paper investigates primitive varieties of general type with maximal irregularity, bounding their bicanonical map degree, and establishing conditions under which the Euler number is one and the bicanonical system separates points.

Contribution

It provides bounds on the degree of bicanonical maps for primitive varieties with maximal irregularity and characterizes the Euler number and point separation in the case of simple Albanese varieties.

Findings

Bounded the degree of the bicanonical map.

Proved that the Euler number is 1 when the Albanese variety is simple.

Showed that |2K_X| separates points over the same general Albanese image.

Abstract

In this note we studied the primitive varieties of general type with $q(X) = dim(X)$ and non-birational bicanonical maps. Let $X$ be such a variety. We bounded the degree of its bicanonical map. If moreover the Albanese variety $Alb(X)$ is simple, we proved that the Euler number $\chi(\omega_X) = 1$, and $|2K_X|$ separates the points mapped to the same general point via the Albanese map.