Twistor lines on cubic surfaces
  It is shown that there exist non-singular cubic surfaces in CP^3 containing 5 twistor lines. This is the maximum number of twistor fibres that a non-singular cubic can contain. Cubic surfaces in CP^3 with 5 twistor lines are classified up to transformations preserving the conformal structure of S^4. 
John Armstrong, Massimiliano Povero, Simon Salamon
math.DG math.AG