On the moduli space of superminimal surfaces in spheres

Using a birational correspondence between the twistor space ofS2nand projective space, we describe, up to birational equivalence, the moduli space of superminimal surfaces inS2nof degreedas curves of degreedin projective space satisfying a certain differential system. Using this approach, we show that the moduli space of linearly full maps is nonempty for sufficiently large degree and we show that the dimension of this moduli space forn=3and genus0is greater than or equal to2d+9. We also give a direct, simple proof of the connectedness of the moduli space of superminimal surfaces inS2nof degreed.