For an irreducible projective variety X, we study the family of h-planes contained in the secant variety S k X , for 0`h`k. These families have an expected dimension and we study varieties for which the expected dimension is not attained; for these varieties, making general consecutive projections to lower dimensional spaces, we do not get the expected singularities. In particular, we examine the family G 1Y 2 of lines sitting in 3-secant planes to a surface S. We show that the actual dimension of G 1Y 2 is equal to the expected dimension unless S is a cone or a rational normal scroll of degree 4 in P 5 . 1991 Mathematics Subject Classi®cation: 14N05. IntroductionLet X r P r be a smooth, non degenerate n-dimensional projective variety. Take P e P r and assume that for all lines L through P the intersection, as a scheme, has length at most one. Then using the projection with center P one obtains an embedding X r P rÀ1 . Such a point P exists if and only if S 1 X H P r , with S 1 X being the secant variety of X: it is the closure of lines spanned by pairs of distinct points in X (see [JH] lecture 15). Clearly dimS 1 X 2n 1 and minfrY 2n 1g can be considered as the``expected dimension'' of S 1 X . Hence, using such projections, we can embed X in P 2n1 ; we can go further and project X isomorphically in some P m , m`2n 1 if and only if S 1 X has dimension smaller than the expected one.The classi®cation of varieties for which S 1 X has dimension less than the expected value was studied by Severi, Terracini and Scorza (see e.g. [Sc]) for objects of small dimension and it has been recently reconsidered by several authors. Severi found that the Veronese surface is the unique smooth surface in P r , r 5 that can be projected isomorphically to P 4 . In [Z], lower bounds for dimS 1 X are proved and a classi®-cation for varieties attained this lower bound is presented. These varieties can be projected isomorphically to some projective space of dimension much smaller than 2n 1.In general, projecting a variety X r P 2n1 from some disjoint linear subspace p of dimension k b 0, as k increases one expects that points of higher multiplicity must Brought to you by | Rutgers University Authenticated Download Date | 6/5/15 12:22 AM

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