Abstract
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.