Projective Equivalence of Smooth Hypersurfaces via Cyclic Covers

TL;DR

This paper demonstrates that the cyclic cover branched along a smooth hypersurface uniquely determines the hypersurface up to projective transformations, solving a previously posed question.

Contribution

It proves that the cyclic $d$-fold cover of a smooth hypersurface uniquely characterizes the hypersurface up to projective equivalence.

Findings

Cyclic covers fully determine smooth hypersurfaces up to projective equivalence.

Answers a question posed by Huybrechts regarding hypersurface characterization.

Provides a new method for classifying hypersurfaces via their cyclic covers.

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, \S 1.5.6].