Classification of special curves in the space of cubic polynomials

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. Let Perm(λ) be the algebraic curve consisting of those cubic polynomials that admit an orbit of periodmand multiplier λ. We also prove that an irreducible component of Perm(λ) is special if and only if λ = 0.