Projective Lifting Theorem, Complex Reflection Groups, and Quasi-Galois Points for Fermat Varieties
Abstract
We investigate birational automorphisms of hypersurfaces that preserve the projection from a point. Let V ⊂ ℙn+1 be an irreducible hypersurface of degree d ≥ 3 with dim Sing(V) ≤ n−2, and let πP : V dashrightarrow ℙn denote the projection from a point P ∈ ℙn+1 ∖ Sing(V). Assume moreover that if d = 3 then P ∉ V. We prove that any σ∈ Bir(V) satisfying πP ∘ σ= πP extends uniquely to a projective transformation of ℙn+1. By combining the above projective lifting theorem with the theory of complex reflection groups, we investigate Galois and quasi-Galois points for the Fermat variety Fdn ⊂ ℙn+1 of dimension n and degree d. We show that, for d ≥ 4, Fdn has exactly n+2 Galois points and, in addition, d(n+2)(n+1)/2 quasi-Galois points. When d=3, F3n has exactly n+2 outer Galois points.
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BibTeX
@misc{miura2026projective,
title = {{Projective Lifting Theorem, Complex Reflection Groups, and Quasi-Galois Points for Fermat Varieties}},
author = {Kei Miura and Takeshi Takahashi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09331},
eprint = {2610.09331},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09331},
}
RIS
TY - UNPB
TI - Projective Lifting Theorem, Complex Reflection Groups, and Quasi-Galois Points for Fermat Varieties
AU - Kei Miura
AU - Takeshi Takahashi
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09331
UR - https://arxiv.org/abs/2610.09331
AB - We investigate birational automorphisms of hypersurfaces that preserve the projection from a point. Let V ⊂ ℙ^n+1 be an irreducible hypersurface of degree d ≥ 3 with dim Sing(V) ≤ n−2, and let π_P : V dashrightarrow ℙ^n denote the projection from a point P ∈ ℙ^n+1 ∖ Sing(V). Assume moreover that if d = 3 then P ∉ V. We prove that any σ∈ Bir(V) satisfying π_P ∘ σ= π_P extends uniquely to a projective transformation of ℙ^n+1. By combining the above projective lifting theorem with the theory of complex reflection groups, we investigate Galois and quasi-Galois points for the Fermat variety F_d^n ⊂ ℙ^n+1 of dimension n and degree d. We show that, for d ≥ 4, F_d^n has exactly n+2 Galois points and, in addition, d(n+2)(n+1)/2 quasi-Galois points. When d=3, F_3^n has exactly n+2 outer Galois points.
ER -