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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.

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