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Structure of Gauss maps. (English. Russian original) Zbl 0623.14026
Funct. Anal. Appl. 21, No. 1-3, 32-41 (1987); translation from Funkts. Anal. Prilozh. 21, No. 1, 39-50 (1987).
The classical Gauss map \(\gamma\) : \(X^ n\to G(N,n)\) associates to a point x of a nonsingular projective algebraic variety \(X^ n\subset {\mathbb{P}}^ N\) the point in the Grassmann variety G(N,n) of n-dimensional linear subspaces in \({\mathbb{P}}^ N\) corresponding to the embedded tangent space \(T_{X,x}\) to X at x. Thus the fiber of \(\gamma\) over a point \(L\in G(N,n)\) is the set of points (with multiplicities) at which the embedded tangent space to X coincides with L. Similarly, for each \(n\leq m\leq N-1\) we consider the higher Gauss map \(\gamma_ m\) whose fiber over a point \(L\in G(N,m)\) coincides with the set of points \(x\in X\) such that \(T_{X,x}\subset L^ m\) (i.e. L is tangent to X at x). We study the structure of the maps \(\gamma_ m\) with special reference to the cases \(m=n\) and \(m=N-1\) and consider applications to tangencies, projections, varieties of small codimension etc.

14N05 Projective techniques in algebraic geometry
14M15 Grassmannians, Schubert varieties, flag manifolds
Full Text: DOI
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