  We exploit techniques from classical (real and complex) algebraic geometry
for the study of the standard twistor fibration $\pi:\mathbb{CP}^{3}\to S^{4}$.
We prove three results about the topology of the twistor discriminant locus of
an algebraic surface in $\mathbb{CP}^{3}$. First of all we prove that, with the
exception of two exceptional cases, the real dimension of the twistor
discriminant locus of an algebraic surface is always equal to 2. Secondly we
describe the possible intersections of a general surface with the family of
twistor lines: we find that only 4 configurations are possible and for each of
them we compute the dimension. Lastly we give a decomposition of the twistor
discriminant locus of a given cone in terms of its singular locus and its dual
variety.
