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