The field of moduli of a divisor on a rational curve

Let k be a field with algebraic closure k¯ and D⊂Pk¯1 a reduced, effective divisor of degree n≥3, write kD for the field of moduli of D. A. Marinatto proved that when n is odd, or n=4, D descends to a divisor on PkD1. We analyze completely the problem of when D descends to a divisor on a smooth, projective curve of genus 0 on kD, possibly with no rational points. In particular, we study the remaining cases n≥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.