HOMOGENEOUS PROJECTIVE VARIETIES WITH DEGENERATE SECANTS

Dedicated to Professor Satoshi Arima on the occasion of his 70th birthday Abstract. The secant variety of a projective variety X in P, denoted by Sec X, is dened 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 dened by δ:= 2 dimX +1−dimSec 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 Sec X 6 = P and δ> 8, and the E6-variety is the only homogeneous projective variety with largest secant de-ciency 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