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

✍ Scribed by Satoru Fukasawa; Hajime Kaji

• Publisher: John Wiley and Sons
• Year: 2008
• Tongue: English
• Weight: 118 KB
• Volume: 281
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200610688

✦ Synopsis

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 ℙ^N^ to re‐embed into some projective space ℙ^M^ 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 ℙ^N^ is identically zero; hence the projective variety__X__ re‐embedded in ℙ^M^ 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 ℙ^N^ with suitable degree in positive characteristic is known to satisfy the assumption above. We give some new, other examples of X in ℙ^N^ satisfying the assumption. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)