| Summary: | We determine explicitly the normalized potential, a Weierstrass-type representation, of a superconformal surface in an even-dimensional sphere $S^{2n}$ in terms of certain normal curvatures of the surface. When the Hopf differential is zero the potential embodies a system of first order equations governing the directrix curve of a superminimal surface in the twistor space of the sphere. We construct a birational map from the twistor space of $S^{2n}$ into ${\Bbb C}P^{n(n+1)/2}$. 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 twistor lift of a superminimal surface in $S^{6}$ as long as the surface does not pass through the north pole. Our approach, which is algebro-geometric in nature, accounts in a rather simple way for the aforementioned first order equations, and as a consequence for the particularly interesting class of superminimal almost complex curves in $S^{6}$. It also yields, in a constructive way, that a generic superminimal surface in $S^{6}$ is {\em not} almost complex and can achieve, by the above degree property, arbitrarily large area.
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