Diversity in Parametric Families of Number Fields

Let X be a projective curve defined over Q and t a non-constant Q-rational function on X of degree at least 2. For every integer n pick a point P_n on X such that t(P_n)=n. A result of Dvornicich and Zannier implies that, for large N, among the number fields Q(P_1),...,Q(P_N) there are at least cN/log N distinct, where c>0. We prove that there are at least N/(log N)^{1-c} distinct fields, where c>0.