  Let $k$ be a field with algebraic closure $\bar{k}$ and $D \subset
\mathbb{P}^{1}_{\bar{k}}$ a reduced, effective divisor of degree $n \ge 3$,
write $k(D)$ for the field of moduli of $D$. A. Marinatto proved that when $n$
is odd, or $n = 4$, $D$ descends to a divisor on $\mathbb{P}^{1}_{k(D)}$.
  We analyze completely the problem of when $D$ descends to a divisor on a
smooth, projective curve of genus $0$ on $k(D)$, possibly with no rational
points. In particular, we study the remaining cases $n \ge 6$ even, and we
obtain conceptual proofs of Marinatto's results and of a theorem by B. Huggins
about the field of moduli of hyperelliptic curves.
