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

Hansol Hong; Hyung-Seok Shin

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arXiv · 1608.06726

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

Hansol Hong·Hyung-Seok Shin

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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 $αδ- βγ= d$ up to an equivalence.

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Hansol Hong, Hyung-Seok Shin. 2016-08-24. Counting of Holomorphic Orbi-spheres in $\mathbb{P}^1_{2,2,2,2}$ and Determinant Equation. https://arxiv.org/abs/1608.06726

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