Moduli space of branched superminimal immersions of a compact Riemann surface into S-4

In this paper we describe the moduli spaces of degree d branched superminim al immersions of a compact Riemann surface of genus g into S-4. We prove th at when d greater than or equal to max{2g, g + 2}, such spaces have the str ucture of projectivized fibre products and are path-connected quasi-project ive varieties of dimension 2d - g + 4. This generalizes known results for s paces of harmonic 2-spheres in S-4.