On Kleiman–Piene's question for Gauss maps

[摘要] 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.

[学科分类] 数学（综合）