Even Cone Spherical Metrics: Blow-Up at a Cone Singularity

We study families of spherical metrics on the flat torus E_τ $=$ C/Λ_τ with blow-up behavior at prescribed conical singularities at $0$ and ± p, where the cone angle at $0$ is $6π$, and at ± p is $4π$. We prove that the existence of such a necessarily unique, even family of spherical metrics is completely determined by the geometry of the torus: such a family exists if and only if the Green function $G(z;τ)$ admits a pair of nontrivial critical points ± a. In this case, the cone point p must equal a, and the corresponding monodromy data is ( 2r,2s), where $a=r+sτ.$ An explicit transformation relating this family to the one with a single conical singularity of angle $6π$ at the origin is established in Theorem 1.4. A rigidity result for rhombic tori is proved in Theorem 1.5.