Singular locus on the space of genus 2 curves with decomposable Jacobians

TL;DR

This paper investigates the singular points on the algebraic surface of genus 2 curves with split Jacobians, identifying specific automorphism groups associated with these singularities for various cases.

Contribution

It characterizes the singular locus for n=2 and n=3 on the surface of genus 2 curves with split Jacobians, and explores the potential for a unified parametrization for all n ≥ 7.

Findings

Singular locus for n=2 corresponds to curves with automorphism groups D4 or D6.

For n=3, the singular locus consists of three specific genus 2 curves with automorphism groups D4 or D6.

The birational parametrization used for n=3 may extend to all n ≥ 7 if the surface is rational.

Abstract

We study the singular locus on the algebraic surface $\S_n$ of genus 2 curves with a $(n, n)$-split Jacobian. Such surface was computed by Shaska in \cite{deg3} for $n=3$, and Shaska at al. in \cite{deg5} for $n=5$. We show that the singular locus for $n=2$ is exactly th locus of the curves of automorphism group $D_4$ or $D_6$. For $n=3$ we use a birational parametrization of the surface $\S_3$ discovered in \cite{deg3} to show that the singular locus is a 0-dimensional subvariety consisting exactly of three genus 2 curves (up to isomorphism) which have automorphism group $D_4$ or $D_6$. We further show that the birational parametrization used in $\S_3$ would work for all $n \geq 7$ if $\S_n$ is a rational surface.