The degree of irrationality of most abelian surfaces is 4

2019/11/01 by Olivier Martin, Martin, Olivier · 2 citations
Mathematics · #Algebraic Geometry and Number Theory #Meromorphic and Entire Functions #Advanced Differential Equations and Dynamical Systems

paper · pdf · doi:10.48550/arxiv.1911.00296

Abstract

The degree of irrationality of a smooth projective variety X is the minimal degree of a dominant rational map X\dashrightarrow ℙ^{dim X}. We show that if an abelian surface A over ℂ is such that the image of the intersection pairing Sym²NS(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\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.

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