# The Bolza curve and some orbifold ball quotient surfaces

## TL;DR

This paper investigates orbifold ball quotient surfaces derived from Abelian surfaces related to the Bolza genus 2 curve, identifying their geometric structures and symmetries through explicit equations and configurations.

## Contribution

It identifies the orbifold surfaces as birational transformations of quotients of Abelian surfaces, computes their mirror equations, and constructs explicit configurations of plane curves and conics.

## Key findings

- Identification of $X$ as $ ext{P}(1,3,8)$
- Explicit equations for the mirror $M$
- Construction of orbifold surfaces from conic arrangements

## Abstract

We study Deraux's non arithmetic orbifold ball quotient surfaces obtained as birational transformations of a quotient $X$ of a particular Abelian surface $A$. Using the fact that $A$ is the Jacobian of the Bolza genus $2$ curve, we identify $X$ as the weighted projective plane $\mathbb{P}(1,3,8)$. We compute the equation of the mirror $M$ of the orbifold ball quotient $(X,M)$ and by taking the quotient by an involution, we obtain an orbifold ball quotient surface with mirror birational to an interesting configuration of plane curves of degrees $1,2$ and $3$. We also exhibit an arrangement of four conics in the plane which provides the above-mentioned ball quotient orbifold surfaces.