On Kleiman–Piene's question for Gauss maps

Abstract

We study the product of a Fermat hypersurface $X_0^{p+1}+\dots+X_n^{p+1}=0 \subset \mathbf{P}^n$ with $n \ge 3$ and $\mathbf{P}^1$, embedded in $\mathbf{P}^{2n+1}$ by Segre embedding where $p>0$ is the characteristic of the base field. This smooth variety is nonreflexive and has Gauss map which is an embedding. This gives a negative answer to the following Kleiman–Piene question in any positive characteristic: does the separability of the Gauss map imply reflexivity? The only known smooth examples, which give a negative answer, are given by Kaji in characteristic 2.