The field of moduli of a divisor on a rational curve

TL;DR

This paper investigates when divisors on rational curves can be defined over their field of moduli, extending previous results and providing new insights into the descent problem for various degrees.

Contribution

It completely characterizes the descent of divisors on rational curves for all degrees, including the challenging even degrees greater than 4, and offers conceptual proofs of existing theorems.

Findings

Complete characterization of divisor descent for all degrees n ≥ 3.

Conceptual proofs of Marinatto's results and Huggins' theorem.

Analysis of cases where the divisor's degree is even and ≥ 6.

Abstract

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.