Projective varieties admitting an embedding with Gauss map of rank zero

Satoru Fukasawa, Katsuhisa Furukawa, Hajime Kaji

研究成果: Article

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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).

元の言語English
ページ(範囲)2645-2661
ページ数17
ジャーナルAdvances in Mathematics
224
発行部数6
DOI
出版物ステータスPublished - 2010 8 1

ASJC Scopus subject areas

  • Mathematics(all)

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