Frobenius liftable hypersurfaces

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.