Hyperelliptic genus 3 curves with involutions and a Prym map

TL;DR

This paper characterizes genus 3 hyperelliptic curves with involutions, provides explicit equations, and explores the Prym map's properties, revealing new insights into their geometric structure and associated abelian surfaces.

Contribution

It explicitly characterizes certain hyperelliptic genus 3 curves with involutions and analyzes the Prym map's degree, also constructing families of polarized abelian surfaces with specific duality properties.

Findings

Prym map is 2:1 when fixing an elliptic quotient

Explicit equations for hyperelliptic genus 3 curves with involutions

Constructs families of polarized abelian surfaces with non-isomorphic duals

Abstract

We characterise genus 3 complex smooth hyperelliptic curves that contain two additional involutions as curves that can be build from five points in $\mathbb{P}^1$ with a distinguished triple. We are able to write down explicit equations for the curves and all their quotient curves. We show that, fixing one of the elliptic quotient curve, the Prym map becomes a 2:1 map and therefore the hyperelliptic Klein Prym map, constructed recently by the first author with A. Ortega, is also 2:1 in this case. As a by-product we show an explicit family of $(1, d)$ polarised abelian surfaces (for d > 1), such that any surface in the family satisfying a certain explicit condition is abstractly non-isomorphic to its dual abelian surface.