Track new results across pure and applied mathematics.
Search papers, labs, and topics across Lattice.
This paper establishes a projective lifting theorem for birational automorphisms of hypersurfaces that preserve a projection from a point, proving such automorphisms uniquely extend to projective transformations of the ambient space. Leveraging this, the authors analyze Galois and quasi-Galois points for Fermat varieties, demonstrating a specific count for $d \ge 4$ and identifying outer Galois points for $d=3$. The findings provide a structural understanding of automorphisms and geometric properties of these important algebraic varieties.
Birational automorphisms preserving projection from a point on hypersurfaces uniquely lift to global projective transformations, revealing a fundamental rigidity.
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 $\pi_P : V \dashrightarrow \mathbb{P}^n$ 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 $\sigma \in \mathrm{Bir}(V)$ 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.