Abstract
For any given projective variety Y, we construct a projective variety 
whose general fiber of the Gauss map with reduced scheme structure is isomorphic to Y when the characteristic >0.
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Fukasawa, S. Varieties with non-linear Gauss fibers. Math. Ann. 334, 235–239 (2006). https://doi.org/10.1007/s00208-005-0688-5
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DOI: https://doi.org/10.1007/s00208-005-0688-5