Homogeneous projective varieties with degenerate secants

Abstract

The secant variety of a projective variety X in P, denoted by Sec X, is defined 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 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 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. Introduction The secant variety of a projective variety X in P, denoted by Sec X , is defined 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 dimX + 1− dim Sec X. In 1979, F. L. Zak proved a significant inequality, 3 dimX + 4 ≤ 2 dim P for a smooth, non-degenerate X with SecX 6= P, which had been conjectured by R. Hartshorne [Ht, Conjecture 4.2] (see also [FL], [LV], [Z]). From the viewpoint of Zak’s inequality, projective varieties X which attain the equality, namely Severi varieties, were studied actively, and Zak finally found that there are exactly four Severi varieties (see [FR], [T], [LV], [Z]): It turns out that those varieties are all homogeneous and have δ = 1, 2, 4, 8. For the extremal case of odd dimensional X , in which 3 dimX +5 = 2 dim P, T. Fujita [F] gave a classification for 3-dimensional X and M. Ohno [O] recently gave classifications for 5-dimensional X and for 7dimensional X under a certain condition, where those X of dimension 3,5,7 have δ = 1, 2, 3, respectively. Thus several authors have studied projective varieties X with δ > 0. The purpose of this article is to list the homogeneous projective varieties X with δ > 0 under the assumption that X arise from irreducible representations of complex simple algebraic groups. Zak already obtained a table of those X in case Received by the editors April 9, 1996. 1991 Mathematics Subject Classification. Primary 14M17, 14N05, 17B10, 20G05. c ©1999 American Mathematical Society 533 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use