Projective Equivalence of Smooth Hypersurfaces via Cyclic Covers
Zhiyuan Li, Zhichao Tang
TL;DR
The paper resolves whether the moduli map $\\Psi_d^n: \\mathcal{M}_{d}^n \\to \\mathcal{M}_{d}^{n+1}$, which sends a smooth degree-$d$ hypersurface to its $d$-fold cyclic cover, is injective for all $n\\ge 2$ and $d\\ge 2$ by leveraging the structure of outer Galois points. The authors show that the cyclic cover uniquely determines the hypersurface up to projective equivalence, providing a Torelli-type refinement for cubic surfaces and threefolds and removing the need for $\\mathbb{Z}[\\rho]$-equivariance in this context. The argument combines a normal-form description of hypersurfaces with outer Galois points and a projective-conjugacy analysis, yielding a simple, uniform proof of rigidity. They also discuss extensions and limitations to positive characteristic, where independence of outer Galois points can fail in certain cases, outlining conditions under which the approach remains valid.
Abstract
In this paper, we prove that for any smooth hypersurface $Y$ of degree $d$ in $\mathbb{P}^{n+1}_k$, the cyclic $d$-fold cover $\widetilde{Y} \to \mathbb{P}^{n+1}_k$ branched along $Y$ completely characterizes $Y$ up to projective equivalence. This solves a question asked by Huybrechts in [Huy23, §1.5.6].