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Extension of solvable coverings across horizontal divisors over the integers
Abstract
We study the effect of deleting a horizontal divisor from an arithmetic scheme on its étale fundamental group. Assume that the scheme is regular along the divisor, that each prime component contains a normal crossings point of residue characteristic $2$, and that the generic fibers of the boundary components remain geometrically integral after finite étale base change. We prove that the kernel of the induced surjection on fundamental groups is topologically perfect. In particular, the inclusion induces an isomorphism on maximal prosolvable quotients. We apply this result to projective hyperplane complements and to the extension of representations over complete local coefficient rings.
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BibTeX
@misc{yamaguchi2026extension,
title = {{Extension of solvable coverings across horizontal divisors over the integers}},
author = {Naganori Yamaguchi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02948},
eprint = {2610.02948},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02948},
}
RIS
TY - UNPB
TI - Extension of solvable coverings across horizontal divisors over the integers
AU - Naganori Yamaguchi
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02948
UR - https://arxiv.org/abs/2610.02948
AB - We study the effect of deleting a horizontal divisor from an arithmetic scheme on its étale fundamental group. Assume that the scheme is regular along the divisor, that each prime component contains a normal crossings point of residue characteristic $2$, and that the generic fibers of the boundary components remain geometrically integral after finite étale base change. We prove that the kernel of the induced surjection on fundamental groups is topologically perfect. In particular, the inclusion induces an isomorphism on maximal prosolvable quotients. We apply this result to projective hyperplane complements and to the extension of representations over complete local coefficient rings.
ER -