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

Satoru Fukasawa, Hajime Kaji

    Research output: Contribution to journalArticle

    5 Citations (Scopus)

    Abstract

    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
    Volume281
    Issue number10
    DOIs
    Publication statusPublished - 2008 Oct

    Fingerprint

    Birational Maps
    Gauss Map
    Projective Variety
    Reflexivity
    Smoothness
    Fermat
    Positive Characteristic
    Projective Space
    Algebraically closed
    Hypersurface
    Imply
    Zero

    Keywords

    • Gauss map
    • Generic smoothness
    • Reflexivity

    ASJC Scopus subject areas

    • Mathematics(all)

    Cite this

    Existence of a non-reflexive embedding with birational Gauss map for a projective variety. / Fukasawa, Satoru; Kaji, Hajime.

    In: Mathematische Nachrichten, Vol. 281, No. 10, 10.2008, p. 1412-1417.

    Research output: Contribution to journalArticle

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