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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
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