We study the product of a Fermat hypersurface X-0(p+1) +...+ X-n(p+1) = 0 subset of P-n with n >= 3 and P-1, embedded in P2n+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.
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