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Let X be an n-dimensional nondegenerate (i.e., not contained in a hyperplane) projective manifold in PN over an algebraically closed field k of characteristic 0. Let SecX denote the secant variety of X in PN . It is well known that dim SecX ≤ min{2n + 1, N}. If dim SecX < min{2n + 1, N}, SecX is said to be degenerate (see [L]). If SecX is degenerate, we define the secant defect of X to be the integer δ = 2n+ 1− dim SecX (see [L-V, § 1f] or [Z, Chap. 5, §1]). The Linear Normality Theorem ([Z, Chap. 2, Corollary 2.17]) implies that if SecX is degenerate then dim SecX has a lower bound, that is, dim SecX ≥ (3n+ 2)/2. If equality holds, X is called a Severi variety. Severi varieties were classified finally by F.L. Zak into only four manifolds up to projective equivalence ([Z, Chap. 4, Theorem 4.7]). After Zak’s complete classification of Severi varieties, R. Lazarsfeld and A. Van de Ven proposed the following problem in [L-V, § 1f], observing that the secant defects of Severi varieties are less than or equal to eight. Problem. Do there exist smooth projective varieties with arbitrarily large secant defect? Zak also generalized the class of Severi varieties to a class of manifolds, named Scorza varieties, and classified Scorza varieties [Z, Chap. 6]. However the secant defects of Scorza varieties are also less than or equal to eight, and it is not yet known whether there exists a manifold with δ ≥ 9 (see Examples 1-9 in Section 1 and [L, Example 2.3]). In this article, we propose a new class of projective manifolds with degenerate secant varieties, which class is wider than the class of Scorza varieties, and investigate some properties of this class of manifolds. In particular we show that there is a strong restriction on dimension of the manifold X in this class with large δ, and we give a classification of such manifolds of low dimension.

On degenerate secant varieties whose Gauss maps have the largest images