Counting of Holomorphic Orbi-spheres in $\mathbb{P}^1_{2,2,2,2}$ and Determinant Equation

TL;DR

This paper counts holomorphic orbi-spheres in a specific orbifold by establishing a correspondence with sublattices of a complex lattice, reducing the counting problem to solving a determinant equation.

Contribution

It introduces an explicit correspondence between holomorphic orbi-spheres and sublattices, providing a novel method to count these spheres via lattice enumeration.

Findings

Established a bijection between orbi-spheres and sublattices.

Reduced counting to solutions of a determinant equation.

Provided explicit formulas for enumeration based on lattice solutions.

Abstract

We count the number of holomorphic orbi-spheres in the $\mathbb{Z}_2$-quotient of an elliptic curve. We first prove that there is an explicit correspondence between the holomorphic orbi-spheres and the sublattices of $\mathbb{Z} \oplus \mathbb{Z} \sqrt{-1} (\subset \mathbb{C})$. The problem of counting sublattices of index $d$ then reduces to find the number of integer solutions of the equation $\alpha \delta - \beta \gamma = d$ up to an equivalence.