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Homogeneous projective varieties with degenerate secants. (English) Zbl 0905.14031
Summary: The secant variety of a projective variety \(X\) in \(\mathbb{P}\), denoted by \(\text{Sec }X\), is defined to be the closure of the union of lines in \(\mathbb{P}\) passing through at least two points of \(X\), and the secant deficiency of \(X\) is defined by \(\delta := 2 \dim X + 1 - \dim \text{Sec }X\). We list the homogeneous projective varieties \(X\) with \(\delta > 0\) under the assumption that \(X\) arises from irreducible representations of complex simple algebraic groups. It turns out that there is no homogeneous, non-degenerate, projective variety \(X\) with \(\text{Sec }X \not = \mathbb{P}\) and \(\delta > 8\), and the \(E_{6}\)-variety is the only homogeneous projective variety with largest secant deficiency \(\delta = 8\). This gives a negative answer to a problem posed by R. Lazarsfeld and A. Van de Ven [“Topics in the geometry of projective spaces. Recent work of F. L. Zak”, DMV Seminar, Vol. 4 (1984; Zbl 0564.14007); §1f] if we restrict ourselves to homogeneous projective varieties.

14M17 Homogeneous spaces and generalizations
14N05 Projective techniques in algebraic geometry
20G05 Representation theory for linear algebraic groups
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