On separable higher Gauss maps

2017/02/20 by Katsuhisa Furukawa, Atsushi Ito, Furukawa, Katsuhisa +1
Mathematics · #14N05 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.1702.06010

openalex publication_date 2017/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the m-th Gauss map in the sense of F.~L.~Zak of a projective variety X ⊂ ℙ^N over an algebraically closed field in any characteristic. For all integer m with n:=dim(X) ≤ m < N, we show that the contact locus on X of a general tangent m-plane is a linear variety if the m-th Gauss map is separable. We also show that for smooth X with n < N-2, the (n+1)-th Gauss map is birational if it is separable, unless X is the Segre embedding ℙ¹ × ℙⁿ ⊂ ℙ^2n-1. This is related to L. Ein's classification of varieties with small dual varieties in characteristic zero.

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