Homogeneous projective varieties with degenerate secants

The secant variety of a projective variety X in ℙ, denoted by SecX, is denned to be the closure of the union of lines in ℙ passing through at least two points of X, and the secant deficiency of X is defined by δ := 2 dim X + 1 -dim Sec X. We list the homogeneous projective varieties X with δ > 0 under the assumption that X arise from irreducible representations of complex simple algebraic groups. It turns out that there is no homogeneous, non-degenerate, projective variety X with SecX 5 P and δ > 8, and the E₆-variety is the only homogeneous projective variety with largest secant deficiency δ = 8. This gives a negative answer to a problem posed by R. Lazarsfeld and A. Van de Ven if we restrict ourselves to homogeneous projective varieties.