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

### 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.Comment: 26 pages, 5 figures, comments welcom

Topics: Mathematics - Symplectic Geometry, 53D45, 57R18
Year: 2016
OAI identifier: oai:arXiv.org:1608.06726

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