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 $V\subset {\mathbb{P}}^{n+1}$ be an irreducible hypersurface of degree $d\ge 3$ with $\dim \mathrm{Sing}(V)\le n-2$, and let ${\pi }_P:V⇢{\mathbb{P}}^n$ 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 $\sigma \in \mathrm{Bir}(V)$ satisfying ${\pi }_P\circ \sigma ={\pi }_P$ 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.

arXiv comment11 pages