Frobenius liftable hypersurfaces

TL;DR

This paper proves that Frobenius liftable hypersurfaces in projective space are necessarily toric divisors, and characterizes certain varieties with toric pairs as projective spaces with toric divisors.

Contribution

It establishes a link between Frobenius liftability modulo $p^2$ and the toric nature of divisors, providing a new characterization of toric divisors in positive characteristic.

Findings

Frobenius liftability implies divisors are toric.

Characterization of varieties with toric pairs as projective spaces.

Connection between Frobenius liftability and toric geometry.

Abstract

Let $D$ be a reduced divisor in $\mathbb P^n_k$ for an algebraically closed field $k$ of positive characteristic $p > 0$. We prove that if $(\mathbb P^n_k, D)$ is Frobenius liftable modulo $p^2$, then $D$ is a toric divisor. As a corollary, we show that if there exists a finite surjective morphism $f\colon Y\to X$ onto a smooth projective complex variety $X$ of Picard rank $1$ such that $(Y, f^{-1}(D)_{\mathrm{red}})$ is a toric pair, then $X$ is the projective space and $D$ is a toric divisor.