Fine points of the moduli space of cubic rational maps

Source abstract

A rational point of the moduli space $\mathcal{M}_3$ of cubic rational maps is fine if its conjugacy class contains a map defined over $\mathbb{Q}$, and coarse otherwise. In the invariants $ξ_0,\dots,ξ_5$ that embed $\mathcal{M}_3$ in $\mathbb{P}(2,2,3,3,4,6)$, we prove that a point with trivial automorphism group is fine if and only if the quaternion algebra $(-m_{12},-2ξ_0)_{\mathbb{Q}}$ splits, where $m_{12}=\frac{1}{3}ξ_0^2ξ_1+ξ_0ξ_4-ξ_3^2$; we give the analogous criterion on the locus $\mathcal{L}_2$ of maps with an involution and show that all other strata are fine. The proof is constructive and yields an explicit descent. We prove that the obstruction class is unramified along every divisor of the compactification, that a prime $p\ge 5$ obstructs only if the point reduces modulo $p$ into the closure of one of the two codimension-two loci $\mathcal{L}_1$, $\mathcal{L}_2$ of maps with an involution, and that $\mathcal{M}_3$ has transverse $A_1$-singularities along both, above which the class ramifies; local densities predict that $\mathcal{L}_2$ is met modulo $p$ with probability $\asymp 1/p$, so that it carries the obstruction. A census of all 214,260,288 rational points of weighted moduli height at most 7 agrees with this mechanism and supports the conjecture that the proportion of fine points tends to zero like $(\log B)^{-1/2}$.