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

Provided by:
arXiv.org e-Print Archive

Downloaded from
http://arxiv.org/abs/1608.06726

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.