2023/09/05 by Nils Bruin, Bruin, Nils, Avinash Kulkarni +1 · 1 citation
Mathematics · #11G10 #14H10 #14H40 #14H45 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2309.01959
openalex publication_date 2023/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the question of when a Jacobian of a curve of genus 2g admits a (2,2)-isogeny to two polarized dimension g abelian varieties. We find that one of them must be a Jacobian itself and, if the associated curve is hyperelliptic, so is the other. For g=2 this allows us to describe (2,2)-decomposable genus 4 Jacobians in terms of Prym varieties. We describe the locus of such genus 4 curves in terms of the geometry of the Igusa quartic threefold. We also explain how our characterization relates to Prym varieties of unramified double covers of plane quartic curves, and we describe this Prym map in terms of 6 and 7 points in ℙ3. We also investigate which genus 4 Jacobians admit a 2-isogeny to the square of a genus 2 Jacobian and give a full description of the hyperelliptic ones. While most of the families we find are of the expected dimension 1, we also find a family of unexpectedly high dimension~2.