  We describe all special curves in the parameter space of complex cubic
polynomials, that is all algebraic irreducible curves containing infinitely
many post-critically finite polynomials. This solves in a strong form a
conjecture by Baker and DeMarco for cubic polynomials.
  We also prove that an irreducible component of the algebraic curve consisting
of those cubic polynomials that admit an orbit of any given period and given
multiplier is special if and only if the multiplier is 0.
