Curves with prescribed rational points

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