On the moduli space of superminimal surfaces in spheres

Abstract

Using a birational correspondence between the twistor space of S 2 n and projective space, we describe, up to birational equivalence, the moduli space of superminimal surfaces in S 2 n of degree d as curves of degree d in 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 for n = 3 and genus 0 is greater than or equal to 2 d + 9. We also give a direct, simple proof of the connectedness of the moduli space of superminimal surfaces in S 2 n of degree d .