The field of moduli of a divisor on a rational curve

Let $k$ be a field with algebraic closure $\bar{k}$ and $D\subset {\mathbb{P}}_{\bar{k}}^1$ 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 ${\mathbb{P}}_{k(D)}^1$. 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.