Hyperbolic 2-spheres with cone singularities

We study the space \(C(a_0,a_1,\dots,a_n)\) of hyperbolic 2-spheres with cone points of prescribed apex curvatures \(2a_0,2a_1,\dots,2a_n\in]0,2\pi[\) and some related spaces. For \(n=3\), we get a detailed description of such spaces. The euclidean 2-spheres were considered by W. P. Thurston: for \(n=4\), the corresponding spaces provide the famous 7 examples of nonarithmetic compact holomorphic 2-ball quotients previously constructed by Deligne-Mostow.