On degenerate secant varieties whose Gauss maps have the largest images
type: Item
instance of: scholarly article
title: On degenerate secant varieties whose Gauss maps have the largest images
zbMATH Open document ID: 0939.14029
author: Masahiro Ohno
published in: Pacific Journal of Mathematics
publication date: 1999-10-11T00:00:00Z
review text: 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. \textit{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.
Mathematics Subject Classification ID: 14N15
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 1344280
zbMATH Keywords: degenerate secant varieties
zbMATH Keywords: Severi variety
zbMATH Keywords: Gauss map
zbMATH Keywords: Fano manifold
reviewed by: Heinrich Guggenheimer
MaRDI profile type: Publication
full work available at URL: pjm.1999.187.151
OpenAlex ID: W2009779049
DOI: 10.2140/PJM.1999.187.151
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description: scientific article; zbMATH DE number 1344280