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On degenerate secant varieties whose Gauss maps have the largest images. (English) Zbl 0939.14029
Let $$X$$ be an $$n$$-dimensional nondegenerate projective manifold in $$\mathbb{P}^N$$ over a field of characteristic zero, Sec $$X$$ the secant variety of $$X$$ in $$\mathbb{P}^N$$. The secant variety is degenerate if $$\text{dim Sec } X<\min (2n+1,N)$$. Then $$\text{dim Sec } X\geq(3n+2)/2$$. If equality holds then $$X$$ is a Severi variety; these have been classified completely by Zak.
F. L. Zak [“Tangents and secants of algebraic varieties”, Transl. Math. Monographs 127 (1993; Zbl 0795.14018)] studied a larger class of manifolds with degenerate secant variety which he called Scorza varieties.
The present author defines a still larger class by studying the Gauss map $$\gamma$$ of the smooth part of Sec $$X$$ and requiring that $$\text{dim Image} (\gamma)=2 (\text{dim Sec } X-n-1)$$. Let $$\varepsilon=2 \text{dim Sec }X-3n-2$$. The author proves many results about these manifolds; among these are that $$\text{dim Sec } X\leq 2n-2$$ implies that $$X$$ is a Fano manifold. He also determines all possible $$n$$ for $$\varepsilon=2, 3, 4, 5$$, and classifies $$X$$ in case $$\text{dim Sec } X=2n-1$$, $$n=6$$, and $$\text{dim Sec }X=2n-2$$, $$n=4,5$$. The proofs need the heavy machinery developed by earlier researchers in this field. The author notes that there is no known example of all these cases that is not a Fano manifold.

MSC:
 14N15 Classical problems, Schubert calculus 14M15 Grassmannians, Schubert varieties, flag manifolds
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