Projective Lifting Theorem, Complex Reflection Groups, and Quasi-Galois Points for Fermat Varieties

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 po