Fine points of the moduli space of cubic rational maps
Abstract
A rational point of the moduli space ℳ3 of cubic rational maps is fine if its conjugacy class contains a map defined over ℚ, and coarse otherwise. In the invariants ξ0,…,ξ5 that embed ℳ3 in 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 (−m12,−2ξ0)ℚ splits, where m12=1/3ξ02ξ1+ξ0ξ4−ξ32; we give the analogous criterion on the locus ℒ2 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≥ 5 obstructs only if the point reduces modulo p into the closure of one of the two codimension-two loci ℒ1, ℒ2 of maps with an involution, and that ℳ3 has transverse A1-singularities along both, above which the class ramifies; local densities predict that ℒ2 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.
Related
- Ideal Classes of Order 2n in a Family of Totally Real Cubic Fields and Rational Points on Associated Elliptic Curves
- Density of repelling periodic points for cubic rational maps
- Hierarchical hyperbolicity, ball quotients, and the moduli space of smooth cubic surfaces
- The moduli space of cubic surfaces and its orbifold fundamental group
- K-Moduli Wall Crossing and Automorphic Forms for the Moduli Space of Rational Elliptic Surfaces
BibTeX
@misc{shaska2026fine,
title = {{Fine points of the moduli space of cubic rational maps}},
author = {Tanush Shaska},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33990},
eprint = {2610.33990},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33990},
}
RIS
TY - UNPB
TI - Fine points of the moduli space of cubic rational maps
AU - Tanush Shaska
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.33990
UR - https://arxiv.org/abs/2610.33990
AB - A rational point of the moduli space ℳ_3 of cubic rational maps is fine if its conjugacy class contains a map defined over ℚ, and coarse otherwise. In the invariants ξ_0,…,ξ_5 that embed ℳ_3 in 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)_ℚ splits, where m_12=1/3ξ_0^2ξ_1+ξ_0ξ_4−ξ_3^2; we give the analogous criterion on the locus 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≥ 5 obstructs only if the point reduces modulo p into the closure of one of the two codimension-two loci ℒ_1, ℒ_2 of maps with an involution, and that ℳ_3 has transverse A_1-singularities along both, above which the class ramifies; local densities predict that 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.
ER -