The degree of irrationality of most abelian surfaces is 4

Abstract

The degree of irrationality of a smooth projective variety X is the minimal degree of a dominant rational map X P dim X .We show that if an abelian surface A over C is such that the image of the intersection pairing Sym 2 N S(A) → 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 ∤ 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.

Topics & Concepts