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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.

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