An example of Schwarz map of reducible hypergeometric equation $E_2$ in two variables

TL;DR

This paper explores the Schwarz map of a reducible hypergeometric system $E_2$ in two variables, revealing its geometric interpretation as a universal Abel-Jacobi map for genus 2 curves.

Contribution

It demonstrates that the Schwarz map of the reducible hypergeometric system $E_2$ can be understood geometrically as a universal Abel-Jacobi map, linking hypergeometric functions to algebraic geometry.

Findings

Schwarz map admits geometric interpretation as Abel-Jacobi map

The hypergeometric system $E_2$ is reducible and rank four

Connection between hypergeometric functions and genus 2 curves

Abstract

We study an Appell hypergeometric system $E_2$ of rank four which is reducible and show that its Schwarz map admits geometric interpretations: the map can be considered as the universal Abel-Jacobi map of a $1$-dimensional family of curves of genus 2.