Projective varieties admitting an embedding with Gauss map of rank zero

Satoru Fukasawa*, Katsuhisa Furukawa, Hajime Kaji

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


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
Issue number6
Publication statusPublished - 2010 Aug


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

ASJC Scopus subject areas

  • Mathematics(all)


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