  We study one-dimensional algebraic families of pairs given by a polynomial
with a marked point. We prove an "unlikely intersection" statement for such
pairs thereby exhibiting strong rigidity features for these pairs. We infer
from this result the dynamical Andr\'e-Oort conjecture for curves in the moduli
space of polynomials, by describing one-dimensional families in this parameter
space containing infinitely many post-critically finite parameters.
