Curves with prescribed rational points.

Given a smooth curve C / Q with genus ≥ 2 , we know by Faltings' Theorem that C ( Q ) is finite. Here we ask the reverse question: given a finite set of rational points S ⊆ P n ( Q ) , does there exist a smooth projective curve C / Q contained in P n such that C ( Q ) = S ? We answer this question in the affirmative by providing an effective algorithm for constructing such a curve.