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Seven ordered points and the nonexistence of arithmetic ball quotient compactifications
Abstract
We prove that the ordered GIT quotient of seven points on ℙ1 with equal weights is not isomorphic to the Baily--Borel compactification of any arithmetic complex ball quotient. Together with our companion paper and the known positive examples, this determines the equal weight case for every n≥6: such a realization exists precisely for n=6,8. The lattice and the algebraic isomorphism are unrestricted, and torsion and divisorial ramification are allowed. The proof combines arithmetic reductions with identities involving Chern classes of fixed subvarieties and intersection inequalities for nef divisors on the GIT quotient. The final comparison includes all fixed components and requires no bound on the degree of an individual fixed curve.
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BibTeX
@misc{nakano2026seven,
title = {{Seven ordered points and the nonexistence of arithmetic ball quotient compactifications}},
author = {Ryunosuke Nakano and Hisato Matsukawa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23227},
eprint = {2610.23227},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23227},
}
RIS
TY - UNPB
TI - Seven ordered points and the nonexistence of arithmetic ball quotient compactifications
AU - Ryunosuke Nakano
AU - Hisato Matsukawa
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.23227
UR - https://arxiv.org/abs/2610.23227
AB - We prove that the ordered GIT quotient of seven points on ℙ^1 with equal weights is not isomorphic to the Baily--Borel compactification of any arithmetic complex ball quotient. Together with our companion paper and the known positive examples, this determines the equal weight case for every n≥6: such a realization exists precisely for n=6,8. The lattice and the algebraic isomorphism are unrestricted, and torsion and divisorial ramification are allowed. The proof combines arithmetic reductions with identities involving Chern classes of fixed subvarieties and intersection inequalities for nef divisors on the GIT quotient. The final comparison includes all fixed components and requires no bound on the degree of an individual fixed curve.
ER -