Consider a holomorphic curve $f:\Sigma\to\mathbb{CP}^3$ of degree, say, $d$ from some Riemann surface to projective space. The Penrose twistor fibration $\pi:\mathbb{CP}^3\to\text{S}^4$ then allows us to define a map $\Gamma:\Sigma\to\text{S}^4$ by $\Gamma=\pi\circ f$.
What geometric properties is such a map $\Gamma$ allowed to have? Also, given a set of points $x_1,\ldots,x_n\in\text{S}^4$ on the four-sphere, is it possible to construct a $\Gamma$ which passes through all of them?