发表机构
Yamaguchi University; Niigata University（山口大学; 新潟大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明保持点投影的超曲面双有理自同构可唯一提升为射影变换，并利用复反射群理论确定了Fermat簇的Galois点与拟Galois点的精确数量。
AI 中文摘要
我们研究保持从一点投影的超曲面的双有理自同构。设 $V \subset \mathbb{P}^{n+1}$ 为次数 $d \ge 3$ 的不可约超曲面，满足 $\dim \mathrm{Sing}(V) \le n-2$，并设 $\pi_P: V \dashrightarrow \mathbb{P}^n$ 表示从点 $P \in \mathbb{P}^{n+1} \setminus \mathrm{Sing}(V)$ 的投影。进一步假设，若 $d = 3$ 则 $P \notin V$。我们证明，任何满足 $\pi_P \circ \sigma = \pi_P$ 的 $\sigma \in \mathrm{Bir}(V)$ 都唯一地延拓为 $\mathbb{P}^{n+1}$ 的射影变换。通过将上述投影提升定理与复反射群理论相结合，我们研究了Fermat簇 $F_d^n \subset \mathbb P^{n+1}$（维数为 $n$，次数为 $d$）的Galois点和拟Galois点。我们证明，对于 $d \ge 4$，$F_d^n$ 恰好有 $n+2$ 个Galois点，此外还有 $d(n+2)(n+1)/2$ 个拟Galois点。当 $d=3$ 时，$F_3^n$ 恰好有 $n+2$ 个外部Galois点。
英文摘要
We investigate birational automorphisms of hypersurfaces that preserve the projection from a point. Let $V \subset \mathbb{P}^{n+1}$ be an irreducible hypersurface of degree $d \ge 3$ with $\dim \mathrm{Sing}(V) \le n-2$, and let $π_P : V \dashrightarrow \mathbb{P}^n$ denote the projection from a point $P \in \mathbb{P}^{n+1} \setminus \mathrm{Sing}(V)$. Assume moreover that if $d = 3$ then $P \notin V$. We prove that any $σ\in \mathrm{Bir}(V)$ satisfying $π_P \circ σ= π_P$ extends uniquely to a projective transformation of $\mathbb{P}^{n+1}$. 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 $F_d^n \subset \mathbb P^{n+1}$ of dimension $n$ and degree $d$. We show that, for $d \ge 4$, $F_d^n$ 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.
Comments11 pages
