Normalized potentials of minimal surfaces in spheres

We determine explicitly the normalized potential, a Weierstrass-type repres entation, of a superconformal surface in an even-dimensional sphere S-2n in terms of certain normal curvatures of the surface. When the Hopf different ial is zero the potential embodies a system of first order equations govern ing the directrix curve of a superminimal surface in the twister space of t he sphere. We construct a birational map from the twister space of S-2n int o Cpn(n+1)/2. In general, birational geometry does not preserve the degree of an algebraic curve. However, we prove that the birational map preserves the degree, up to a factor 2, of the twister lift of a superminimal surface in S-6 as long as the surface does not pass through the north pole. Our ap proach, which is algebro-geometric in nature, accounts in a rather simple w ay for the aforementioned first order equations, and as a consequence for t he particularly interesting class of superminimal almost complex curves in S-6. It also yields, in a constructive way, that a generic superminimal sur face in S-6 is not almost complex and can achieve, by the above degree prop erty, arbitrarily large area.