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Existence of a non‐reflexive embedding with birational Gauss map for a projective variety
Author(s) -
Fukasawa Satoru,
Kaji Hajime
Publication year - 2008
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200610688
Subject(s) - mathematics , projective variety , variety (cybernetics) , projective space , gauss , pure mathematics , gauss map , twisted cubic , projective test , smoothness , algebraically closed field , embedding , discrete mathematics , complex projective space , mathematical analysis , computer science , statistics , physics , quantum mechanics , artificial intelligence
We study the relationship between the generic smoothness of the Gauss map and the reflexivity (with respect to the projective dual) for a projective variety defined over an algebraically closed field. The problem we discuss here is whether it is possible for a projective variety X in ℙ N to re‐embed into some projective space ℙ M so as to be non‐reflexive with generically smooth Gauss map. Our result is that the answer is affirmative under the assumption that X has dimension at least 3 and the differential of the Gauss map of X in ℙ N is identically zero; hence the projective variety X re‐embedded in ℙ M yields a negative answer to Kleiman–Piene's question: Does the generic smoothness of the Gauss map imply reflexivity for a projective variety? A Fermat hypersurface in ℙ N with suitable degree in positive characteristic is known to satisfy the assumption above. We give some new, other examples of X in ℙ N satisfying the assumption. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)