  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.
