Moduli of spherical tori with one conical point

We determine the topology of the moduli space MS1;1(v) of surfaces of genus one with a Riemannian metric of constant curvature 1 and one conical point of angle 2 pi v. In particular, for v is an element of (2m-1, 2m + 1) nonodd, MS1,1(v) is connected, has orbifold Euler characteristic -1/12m(2), and its topology depends on the integer m > 0 only. For v= 2m + 1 odd, MS1,1(v) has 1/6 m(m + 1) connected components. For v= 2m even, MS1,1(v) has a natural complex structure and it is biholomorphic to H-2/G(m) for a certain subgroup Gm of SL(2, Z) of index m(2), which is nonnormal for m > 1.