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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.

##### MSC:
 14N05 Projective techniques in algebraic geometry 14M15 Grassmannians, Schubert varieties, flag manifolds
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