Existence of a non-reflexive embedding with birational Gauss map for a projective variety

Satoru Fukasawa, Hajime Kaji

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5 Citations (Scopus)


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 ℙN to re-embed into some projective space ℙM 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 ℙN is identically zero; hence the projective variety X re-embedded in ℙM 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 ℙN with suitable degree in positive characteristic is known to satisfy the assumption above. We give some new, other examples of X in ℙN satisfying the assumption.

Original languageEnglish
Pages (from-to)1412-1417
Number of pages6
JournalMathematische Nachrichten
Issue number10
Publication statusPublished - 2008 Oct 1



  • Gauss map
  • Generic smoothness
  • Reflexivity

ASJC Scopus subject areas

  • Mathematics(all)

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