Projective varieties admitting an embedding with Gauss map of rank zero

Satoru Fukasawa, Katsuhisa Furukawa, Hajime Kaji

    Research output: Contribution to journalArticle

    1 Citation (Scopus)

    Abstract

    We introduce an intrinsic property for a projective variety as follows: there exists an embedding into some projective space such that the Gauss map is of rank zero, which we call (GMRZ) for short. It turns out that (GMRZ) imposes strong restrictions on rational curves on projective varieties: In fact, using (GMRZ), we show that, contrary to the characteristic zero case, the existence of free rational curves does not imply that of minimal free rational curves in positive characteristic case. We also focus attention on Segre varieties, Grassmann varieties, and hypersurfaces of low degree. In particular, we give a characterisation of Fermat cubic hypersurfaces in terms of (GMRZ), and show that a general hypersurface of low degree does not satisfy (GMRZ).

    Original languageEnglish
    Pages (from-to)2645-2661
    Number of pages17
    JournalAdvances in Mathematics
    Volume224
    Issue number6
    DOIs
    Publication statusPublished - 2010 Aug

    Fingerprint

    Gauss Map
    Rational Curves
    Projective Variety
    Hypersurface
    Zero
    Segre Variety
    Fermat
    Positive Characteristic
    Projective Space
    Restriction
    Imply

    Keywords

    • Gauss map
    • Hypersurface
    • Inseparable
    • Minimal free rational curve
    • Normal bundle

    ASJC Scopus subject areas

    • Mathematics(all)

    Cite this

    Projective varieties admitting an embedding with Gauss map of rank zero. / Fukasawa, Satoru; Furukawa, Katsuhisa; Kaji, Hajime.

    In: Advances in Mathematics, Vol. 224, No. 6, 08.2010, p. 2645-2661.

    Research output: Contribution to journalArticle

    Fukasawa, Satoru ; Furukawa, Katsuhisa ; Kaji, Hajime. / Projective varieties admitting an embedding with Gauss map of rank zero. In: Advances in Mathematics. 2010 ; Vol. 224, No. 6. pp. 2645-2661.
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