Math. Ann. 313, 507–545 (1999) Mathematische Annalen c  Springer-Verlag 1999 Projective invariants of quadratic embeddings F. L. Zak 1,2 1 Central Economics Mathematical Institute of the Russian Academy of Sciences, 47 Nakhi- movskii avenue, Moscow 117418, Russia (e-mail: zak@mccme.ru) 2 Independent University of Moscow, 11 B. Vlas’evski˘ ı, Moscow 121002, Russia Received: 4 May 1998 Mathematics Subject Classification (1991): 14N05, 14M07, 14J40. 0. Introduction Let X ⊂ P r be a nondegenerate projective algebraic variety. It is natural to consider the variety SX ⊃ X swept out by the chords of X , the variety S 2 X ⊃ SX swept out by the trisecant planes of X , and, in general, the variety S k X swept out by the k-dimensional linear subspaces of P r that are (k + 1)-secant to X . In this way one gets an ascending filtration X ⊂ SX ⊂ S 2 X ⊂···⊂ S k X ⊂···⊂ S k 1 −1 X ⊂ S k 1 X = P r , where k 1 = k 1 (X ) = min {k   S k X = P k } and S k−1 X = S k X for 1 ≤ k ≤ k 1 . For example, if r =(a + 1) × (b + 1) − 1, P r is the projectivization of the linear space of (a +1) × (b +1)-matrices, and X = P a × P b ⊂ P r is the Segre variety corresponding to the (a + 1) × (b + 1)-matrices of rank one, then S k X corresponds to the matrices whose rank does not exceed k +1 and k 1 (X ) = min {a, b}. The number k 1 and the dimensions of the secant varieties s k (X )= dim S k (X ) yield a natural collection of invariants of the projective variety X . There are, however, two serious drawbacks: This work was completed while the author was partially supported by RFBR grant 98-01- 01041 and INTAS grant 96-807