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On separable higher Gauss maps. (English) Zbl 1427.14112
Let $$X\subset \mathbb {P}^N$$ be an integral and non-degenerate variety defined over an algebraically closed field. Let $$X^\ast$$ denote the dual variety of $$X$$. For any integer $$m$$ such that $$n\leq m<N$$ let $$G(m,N)$$ be the Grassmannian of all $$m$$-dimensional linear subspace of $$\mathbb {P}^N$$. The $$m$$-th Gauss map $$\gamma _m$$ of $$X$$ (in the sense of Zak) is the rational map $$X\dashrightarrow G(m,N)$$ sending each $$p\in X_{\mathrm{reg}}$$ to the set of all $$A\in G(m,N)$$ containing $$T_pX$$. The authors proves that the general fiber of $$\gamma _m$$ is a linear space if $$\gamma _m$$ is separable. Then (without assuming the separability of $$\gamma _m$$) they study the $$m$$-defect $$\delta _m:= \dim \gamma _m(X_{\mathrm{reg}}) -\dim X^\ast$$. If $$N\ge n+3$$, $$X$$ is smooth and $$\delta_{n+1}$$ is separable, they prove that the following are equivalent:
(1) $$\delta _{n+i}=0$$ for an integer $$i$$ such that $$0<i<N-n-1$$;
(2) $$\delta _{n+i}=0$$ for all integers $$i$$ such that $$0\leq i\leq N-n-1$$;
(3) $$X$$ is the image of the Segre embedding $$\mathbb {P}^1\times \mathbb {P}^{n-1} \to \mathbb {P}^{2n-1}$$.
This theorem is related Ein’s classification of smooth variety with high defect, i.e. with low dimensional $$X^\ast$$.
##### MSC:
 14N05 Projective techniques in algebraic geometry 14J40 $$n$$-folds ($$n>4$$) 14M15 Grassmannians, Schubert varieties, flag manifolds 14M07 Low codimension problems in algebraic geometry
##### Keywords:
Gauss map; dual variety; higher Gauss map; reflexivity; defect
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