The degree of irrationality of most abelian surfaces is 4

TL;DR

This paper investigates the degree of irrationality of abelian surfaces, showing that under certain conditions it is exactly 4, providing new examples and insights into their geometric properties.

Contribution

It establishes that many abelian surfaces have degree of irrationality 4, answering longstanding questions and demonstrating that this invariant is not preserved under isogeny.

Findings

Most abelian surfaces have degree of irrationality 4.

Provides first examples of abelian surfaces with irrationality degree greater than 3.

Shows irrationality degree is not isogeny-invariant for abelian surfaces.

Abstract

The degree of irrationality of a smooth projective variety $X$ is the minimal degree of a dominant rational map $X\dashrightarrow \mathbb{P}^{\dim X}$. We show that if an abelian surface $A$ over $\mathbb{C}$ is such that the image of the intersection pairing $\text{Sym}^2NS(A)\to \mathbb{Z}$ does not contain $12$, then it has degree of irrationality $4$. In particular, a very general $(1,d)$-polarized abelian surface has degree of irrationality $4$ provided that $d\nmid 6$. This answers two questions of Yoshihara by providing the first examples of abelian surfaces with degree of irrationality greater than $3$ and showing that the degree of irrationality is not isogeny-invariant for abelian surfaces.