Homogeneous projective varieties with degenerate secants

The secant variety of a projective variety X in P, denoted by Sec X, is def ined 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 delta := 2 di m X + 1 - dim Sec X. Mie list the homogeneous projective varieties X with d elta > 0 under the assumption that X arise from irreducible representations of complex simple algebraic groups. It turns out that there is no homogene ous, non-degenerate, projective variety X with SecX not equal P and delta > 8, and the Es-variety is the only homogeneous projective variety with larg est secant deficiency delta = 8. This gives a negative answer to a problem posed by R. Lazarsfeld and A. Van de Ven if we restrict ourselves to homoge neous projective varieties.